CINXE.COM

<!DOCTYPE html> <html class="client-nojs skin-theme-clientpref-day mf-expand-sections-clientpref-0 mf-font-size-clientpref-small mw-mf-amc-clientpref-0" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Notation in probability and statistics - Wikipedia</title> <script>(function(){var className="client-js skin-theme-clientpref-day mf-expand-sections-clientpref-0 mf-font-size-clientpref-small mw-mf-amc-clientpref-0";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"3436c978-4326-4ca9-9e6c-98cf86583365","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Notation_in_probability_and_statistics","wgTitle":"Notation in probability and statistics","wgCurRevisionId":1256842542, "wgRevisionId":1256842542,"wgArticleId":2142734,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Notation_in_probability_and_statistics","wgRelevantArticleId":2142734,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFMode":"stable","wgMFAmc":false,"wgMFAmcOutreachActive":false,"wgMFAmcOutreachUserEligible":false,"wgMFLazyLoadImages":true,"wgMFEditNoticesFeatureConflict":false,"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby": true},"wgMFIsSupportedEditRequest":true,"wgMFScriptPath":"","wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":true,"wgEditSubmitButtonLabelPublish":true,"wgSectionTranslationMissingLanguages":[{"lang":"ace","autonym":"Acèh","dir":"ltr"},{"lang":"ady","autonym":"адыгабзэ","dir":"ltr"},{"lang":"alt","autonym":"алтай тил","dir":"ltr"},{"lang":"am","autonym":"አማርኛ","dir":"ltr"},{"lang":"ami","autonym":"Pangcah","dir":"ltr"},{"lang":"an","autonym":"aragonés","dir":"ltr"},{"lang":"ang","autonym":"Ænglisc","dir":"ltr"},{"lang":"ann","autonym":"Obolo","dir":"ltr"},{"lang":"anp","autonym":"अंगिका","dir":"ltr"},{"lang":"ar","autonym":"العربية","dir":"rtl"},{"lang":"ary","autonym":"الدارجة","dir":"rtl"},{"lang":"arz","autonym":"مصرى","dir":"rtl"},{"lang":"as","autonym":"অসমীয়া","dir":"ltr"},{"lang":"ast","autonym":"asturianu","dir":"ltr"},{"lang":"av" ,"autonym":"авар","dir":"ltr"},{"lang":"avk","autonym":"Kotava","dir":"ltr"},{"lang":"awa","autonym":"अवधी","dir":"ltr"},{"lang":"ay","autonym":"Aymar aru","dir":"ltr"},{"lang":"az","autonym":"azərbaycanca","dir":"ltr"},{"lang":"azb","autonym":"تۆرکجه","dir":"rtl"},{"lang":"ba","autonym":"башҡортса","dir":"ltr"},{"lang":"ban","autonym":"Basa Bali","dir":"ltr"},{"lang":"bar","autonym":"Boarisch","dir":"ltr"},{"lang":"bbc","autonym":"Batak Toba","dir":"ltr"},{"lang":"bcl","autonym":"Bikol Central","dir":"ltr"},{"lang":"bdr","autonym":"Bajau Sama","dir":"ltr"},{"lang":"be","autonym":"беларуская","dir":"ltr"},{"lang":"bew","autonym":"Betawi","dir":"ltr"},{"lang":"bg","autonym":"български","dir":"ltr"},{"lang":"bho","autonym":"भोजपुरी","dir":"ltr"},{"lang":"bi","autonym":"Bislama","dir":"ltr"},{"lang":"bjn","autonym":"Banjar","dir":"ltr"},{"lang":"blk","autonym":"ပအိုဝ်ႏဘာႏသာႏ","dir":"ltr"},{"lang":"bm", "autonym":"bamanankan","dir":"ltr"},{"lang":"bn","autonym":"বাংলা","dir":"ltr"},{"lang":"bo","autonym":"བོད་ཡིག","dir":"ltr"},{"lang":"bpy","autonym":"বিষ্ণুপ্রিয়া মণিপুরী","dir":"ltr"},{"lang":"br","autonym":"brezhoneg","dir":"ltr"},{"lang":"bs","autonym":"bosanski","dir":"ltr"},{"lang":"btm","autonym":"Batak Mandailing","dir":"ltr"},{"lang":"bug","autonym":"Basa Ugi","dir":"ltr"},{"lang":"ca","autonym":"català","dir":"ltr"},{"lang":"cdo","autonym":"閩東語 / Mìng-dĕ̤ng-ngṳ̄","dir":"ltr"},{"lang":"ce","autonym":"нохчийн","dir":"ltr"},{"lang":"ceb","autonym":"Cebuano","dir":"ltr"},{"lang":"ch","autonym":"Chamoru","dir":"ltr"},{"lang":"chr","autonym":"ᏣᎳᎩ","dir":"ltr"},{"lang":"ckb","autonym":"کوردی","dir":"rtl"},{"lang":"co","autonym":"corsu","dir":"ltr"},{"lang":"cr","autonym":"Nēhiyawēwin / ᓀᐦᐃᔭᐍᐏᐣ","dir":"ltr"},{"lang":"crh","autonym":"qırımtatarca","dir":"ltr"},{"lang":"cs" ,"autonym":"čeština","dir":"ltr"},{"lang":"cu","autonym":"словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ","dir":"ltr"},{"lang":"cy","autonym":"Cymraeg","dir":"ltr"},{"lang":"da","autonym":"dansk","dir":"ltr"},{"lang":"dag","autonym":"dagbanli","dir":"ltr"},{"lang":"de","autonym":"Deutsch","dir":"ltr"},{"lang":"dga","autonym":"Dagaare","dir":"ltr"},{"lang":"din","autonym":"Thuɔŋjäŋ","dir":"ltr"},{"lang":"diq","autonym":"Zazaki","dir":"ltr"},{"lang":"dsb","autonym":"dolnoserbski","dir":"ltr"},{"lang":"dtp","autonym":"Kadazandusun","dir":"ltr"},{"lang":"dv","autonym":"ދިވެހިބަސް","dir":"rtl"},{"lang":"dz","autonym":"ཇོང་ཁ","dir":"ltr"},{"lang":"ee","autonym":"eʋegbe","dir":"ltr"},{"lang":"el","autonym":"Ελληνικά","dir":"ltr"},{"lang":"eml","autonym":"emiliàn e rumagnòl","dir":"ltr"},{"lang":"eo","autonym":"Esperanto","dir":"ltr"},{"lang":"es","autonym":"español","dir":"ltr"},{"lang":"et","autonym":"eesti","dir":"ltr"},{"lang":"eu","autonym": "euskara","dir":"ltr"},{"lang":"fa","autonym":"فارسی","dir":"rtl"},{"lang":"fat","autonym":"mfantse","dir":"ltr"},{"lang":"ff","autonym":"Fulfulde","dir":"ltr"},{"lang":"fi","autonym":"suomi","dir":"ltr"},{"lang":"fj","autonym":"Na Vosa Vakaviti","dir":"ltr"},{"lang":"fo","autonym":"føroyskt","dir":"ltr"},{"lang":"fon","autonym":"fɔ̀ngbè","dir":"ltr"},{"lang":"fr","autonym":"français","dir":"ltr"},{"lang":"frp","autonym":"arpetan","dir":"ltr"},{"lang":"frr","autonym":"Nordfriisk","dir":"ltr"},{"lang":"fur","autonym":"furlan","dir":"ltr"},{"lang":"fy","autonym":"Frysk","dir":"ltr"},{"lang":"gag","autonym":"Gagauz","dir":"ltr"},{"lang":"gan","autonym":"贛語","dir":"ltr"},{"lang":"gcr","autonym":"kriyòl gwiyannen","dir":"ltr"},{"lang":"gl","autonym":"galego","dir":"ltr"},{"lang":"glk","autonym":"گیلکی","dir":"rtl"},{"lang":"gn","autonym":"Avañe'ẽ","dir":"ltr"},{"lang":"gom","autonym":"गोंयची कोंकणी / Gõychi Konknni","dir":"ltr"},{"lang":"gor", "autonym":"Bahasa Hulontalo","dir":"ltr"},{"lang":"gpe","autonym":"Ghanaian Pidgin","dir":"ltr"},{"lang":"gu","autonym":"ગુજરાતી","dir":"ltr"},{"lang":"guc","autonym":"wayuunaiki","dir":"ltr"},{"lang":"gur","autonym":"farefare","dir":"ltr"},{"lang":"guw","autonym":"gungbe","dir":"ltr"},{"lang":"gv","autonym":"Gaelg","dir":"ltr"},{"lang":"ha","autonym":"Hausa","dir":"ltr"},{"lang":"hak","autonym":"客家語 / Hak-kâ-ngî","dir":"ltr"},{"lang":"haw","autonym":"Hawaiʻi","dir":"ltr"},{"lang":"he","autonym":"עברית","dir":"rtl"},{"lang":"hi","autonym":"हिन्दी","dir":"ltr"},{"lang":"hif","autonym":"Fiji Hindi","dir":"ltr"},{"lang":"hr","autonym":"hrvatski","dir":"ltr"},{"lang":"hsb","autonym":"hornjoserbsce","dir":"ltr"},{"lang":"ht","autonym":"Kreyòl ayisyen","dir":"ltr"},{"lang":"hu","autonym":"magyar","dir":"ltr"},{"lang":"hy","autonym":"հայերեն","dir":"ltr"},{"lang":"hyw","autonym":"Արեւմտահայերէն","dir":"ltr"},{"lang":"ia","autonym": "interlingua","dir":"ltr"},{"lang":"iba","autonym":"Jaku Iban","dir":"ltr"},{"lang":"ie","autonym":"Interlingue","dir":"ltr"},{"lang":"ig","autonym":"Igbo","dir":"ltr"},{"lang":"igl","autonym":"Igala","dir":"ltr"},{"lang":"ilo","autonym":"Ilokano","dir":"ltr"},{"lang":"io","autonym":"Ido","dir":"ltr"},{"lang":"is","autonym":"íslenska","dir":"ltr"},{"lang":"it","autonym":"italiano","dir":"ltr"},{"lang":"iu","autonym":"ᐃᓄᒃᑎᑐᑦ / inuktitut","dir":"ltr"},{"lang":"ja","autonym":"日本語","dir":"ltr"},{"lang":"jam","autonym":"Patois","dir":"ltr"},{"lang":"jv","autonym":"Jawa","dir":"ltr"},{"lang":"ka","autonym":"ქართული","dir":"ltr"},{"lang":"kaa","autonym":"Qaraqalpaqsha","dir":"ltr"},{"lang":"kab","autonym":"Taqbaylit","dir":"ltr"},{"lang":"kbd","autonym":"адыгэбзэ","dir":"ltr"},{"lang":"kbp","autonym":"Kabɩyɛ","dir":"ltr"},{"lang":"kcg","autonym":"Tyap","dir":"ltr"},{"lang":"kg","autonym":"Kongo","dir":"ltr"},{"lang":"kge","autonym":"Kumoring","dir": "ltr"},{"lang":"ki","autonym":"Gĩkũyũ","dir":"ltr"},{"lang":"kk","autonym":"қазақша","dir":"ltr"},{"lang":"kl","autonym":"kalaallisut","dir":"ltr"},{"lang":"km","autonym":"ភាសាខ្មែរ","dir":"ltr"},{"lang":"kn","autonym":"ಕನ್ನಡ","dir":"ltr"},{"lang":"ko","autonym":"한국어","dir":"ltr"},{"lang":"koi","autonym":"перем коми","dir":"ltr"},{"lang":"krc","autonym":"къарачай-малкъар","dir":"ltr"},{"lang":"ks","autonym":"कॉशुर / کٲشُر","dir":"rtl"},{"lang":"ku","autonym":"kurdî","dir":"ltr"},{"lang":"kus","autonym":"Kʋsaal","dir":"ltr"},{"lang":"kv","autonym":"коми","dir":"ltr"},{"lang":"kw","autonym":"kernowek","dir":"ltr"},{"lang":"ky","autonym":"кыргызча","dir":"ltr"},{"lang":"lad","autonym":"Ladino","dir":"ltr"},{"lang":"lb","autonym":"Lëtzebuergesch","dir":"ltr"},{"lang":"lez","autonym":"лезги","dir":"ltr"},{"lang":"lg","autonym":"Luganda","dir":"ltr"},{"lang":"li","autonym":"Limburgs","dir": "ltr"},{"lang":"lij","autonym":"Ligure","dir":"ltr"},{"lang":"lld","autonym":"Ladin","dir":"ltr"},{"lang":"lmo","autonym":"lombard","dir":"ltr"},{"lang":"ln","autonym":"lingála","dir":"ltr"},{"lang":"lo","autonym":"ລາວ","dir":"ltr"},{"lang":"lt","autonym":"lietuvių","dir":"ltr"},{"lang":"ltg","autonym":"latgaļu","dir":"ltr"},{"lang":"lv","autonym":"latviešu","dir":"ltr"},{"lang":"mad","autonym":"Madhurâ","dir":"ltr"},{"lang":"mai","autonym":"मैथिली","dir":"ltr"},{"lang":"map-bms","autonym":"Basa Banyumasan","dir":"ltr"},{"lang":"mdf","autonym":"мокшень","dir":"ltr"},{"lang":"mg","autonym":"Malagasy","dir":"ltr"},{"lang":"mhr","autonym":"олык марий","dir":"ltr"},{"lang":"mi","autonym":"Māori","dir":"ltr"},{"lang":"min","autonym":"Minangkabau","dir":"ltr"},{"lang":"mk","autonym":"македонски","dir":"ltr"},{"lang":"ml","autonym":"മലയാളം","dir":"ltr"},{"lang":"mn","autonym":"монгол","dir":"ltr"},{"lang":"mni","autonym": "ꯃꯤꯇꯩ ꯂꯣꯟ","dir":"ltr"},{"lang":"mnw","autonym":"ဘာသာမန်","dir":"ltr"},{"lang":"mos","autonym":"moore","dir":"ltr"},{"lang":"mr","autonym":"मराठी","dir":"ltr"},{"lang":"mrj","autonym":"кырык мары","dir":"ltr"},{"lang":"ms","autonym":"Bahasa Melayu","dir":"ltr"},{"lang":"mt","autonym":"Malti","dir":"ltr"},{"lang":"mwl","autonym":"Mirandés","dir":"ltr"},{"lang":"my","autonym":"မြန်မာဘာသာ","dir":"ltr"},{"lang":"myv","autonym":"эрзянь","dir":"ltr"},{"lang":"mzn","autonym":"مازِرونی","dir":"rtl"},{"lang":"nah","autonym":"Nāhuatl","dir":"ltr"},{"lang":"nan","autonym":"閩南語 / Bân-lâm-gú","dir":"ltr"},{"lang":"nap","autonym":"Napulitano","dir":"ltr"},{"lang":"nb","autonym":"norsk bokmål","dir":"ltr"},{"lang":"nds","autonym":"Plattdüütsch","dir":"ltr"},{"lang":"nds-nl","autonym":"Nedersaksies","dir":"ltr"},{"lang":"ne","autonym":"नेपाली","dir":"ltr"},{"lang":"new","autonym": "नेपाल भाषा","dir":"ltr"},{"lang":"nia","autonym":"Li Niha","dir":"ltr"},{"lang":"nl","autonym":"Nederlands","dir":"ltr"},{"lang":"nn","autonym":"norsk nynorsk","dir":"ltr"},{"lang":"nqo","autonym":"ߒߞߏ","dir":"rtl"},{"lang":"nr","autonym":"isiNdebele seSewula","dir":"ltr"},{"lang":"nso","autonym":"Sesotho sa Leboa","dir":"ltr"},{"lang":"ny","autonym":"Chi-Chewa","dir":"ltr"},{"lang":"oc","autonym":"occitan","dir":"ltr"},{"lang":"om","autonym":"Oromoo","dir":"ltr"},{"lang":"or","autonym":"ଓଡ଼ିଆ","dir":"ltr"},{"lang":"os","autonym":"ирон","dir":"ltr"},{"lang":"pa","autonym":"ਪੰਜਾਬੀ","dir":"ltr"},{"lang":"pag","autonym":"Pangasinan","dir":"ltr"},{"lang":"pam","autonym":"Kapampangan","dir":"ltr"},{"lang":"pap","autonym":"Papiamentu","dir":"ltr"},{"lang":"pcd","autonym":"Picard","dir":"ltr"},{"lang":"pcm","autonym":"Naijá","dir":"ltr"},{"lang":"pdc","autonym":"Deitsch","dir":"ltr"},{"lang":"pl","autonym":"polski","dir":"ltr"},{"lang":"pms", "autonym":"Piemontèis","dir":"ltr"},{"lang":"pnb","autonym":"پنجابی","dir":"rtl"},{"lang":"ps","autonym":"پښتو","dir":"rtl"},{"lang":"pwn","autonym":"pinayuanan","dir":"ltr"},{"lang":"qu","autonym":"Runa Simi","dir":"ltr"},{"lang":"rm","autonym":"rumantsch","dir":"ltr"},{"lang":"rn","autonym":"ikirundi","dir":"ltr"},{"lang":"ro","autonym":"română","dir":"ltr"},{"lang":"rsk","autonym":"руски","dir":"ltr"},{"lang":"rue","autonym":"русиньскый","dir":"ltr"},{"lang":"rup","autonym":"armãneashti","dir":"ltr"},{"lang":"rw","autonym":"Ikinyarwanda","dir":"ltr"},{"lang":"sa","autonym":"संस्कृतम्","dir":"ltr"},{"lang":"sah","autonym":"саха тыла","dir":"ltr"},{"lang":"sat","autonym":"ᱥᱟᱱᱛᱟᱲᱤ","dir":"ltr"},{"lang":"sc","autonym":"sardu","dir":"ltr"},{"lang":"scn","autonym":"sicilianu","dir":"ltr"},{"lang":"sco","autonym":"Scots","dir":"ltr"},{"lang":"sd","autonym":"سنڌي","dir":"rtl"},{"lang":"se","autonym":"davvisámegiella", "dir":"ltr"},{"lang":"sg","autonym":"Sängö","dir":"ltr"},{"lang":"sgs","autonym":"žemaitėška","dir":"ltr"},{"lang":"sh","autonym":"srpskohrvatski / српскохрватски","dir":"ltr"},{"lang":"shi","autonym":"Taclḥit","dir":"ltr"},{"lang":"shn","autonym":"ၽႃႇသႃႇတႆး ","dir":"ltr"},{"lang":"si","autonym":"සිංහල","dir":"ltr"},{"lang":"sk","autonym":"slovenčina","dir":"ltr"},{"lang":"skr","autonym":"سرائیکی","dir":"rtl"},{"lang":"sl","autonym":"slovenščina","dir":"ltr"},{"lang":"sm","autonym":"Gagana Samoa","dir":"ltr"},{"lang":"smn","autonym":"anarâškielâ","dir":"ltr"},{"lang":"sn","autonym":"chiShona","dir":"ltr"},{"lang":"so","autonym":"Soomaaliga","dir":"ltr"},{"lang":"sq","autonym":"shqip","dir":"ltr"},{"lang":"sr","autonym":"српски / srpski","dir":"ltr"},{"lang":"srn","autonym":"Sranantongo","dir":"ltr"},{"lang":"ss","autonym":"SiSwati","dir":"ltr"},{"lang":"st","autonym":"Sesotho","dir":"ltr"},{"lang":"stq","autonym": "Seeltersk","dir":"ltr"},{"lang":"su","autonym":"Sunda","dir":"ltr"},{"lang":"sv","autonym":"svenska","dir":"ltr"},{"lang":"sw","autonym":"Kiswahili","dir":"ltr"},{"lang":"szl","autonym":"ślůnski","dir":"ltr"},{"lang":"ta","autonym":"தமிழ்","dir":"ltr"},{"lang":"tay","autonym":"Tayal","dir":"ltr"},{"lang":"tcy","autonym":"ತುಳು","dir":"ltr"},{"lang":"tdd","autonym":"ᥖᥭᥰ ᥖᥬᥲ ᥑᥨᥒᥰ","dir":"ltr"},{"lang":"te","autonym":"తెలుగు","dir":"ltr"},{"lang":"tet","autonym":"tetun","dir":"ltr"},{"lang":"tg","autonym":"тоҷикӣ","dir":"ltr"},{"lang":"th","autonym":"ไทย","dir":"ltr"},{"lang":"ti","autonym":"ትግርኛ","dir":"ltr"},{"lang":"tk","autonym":"Türkmençe","dir":"ltr"},{"lang":"tl","autonym":"Tagalog","dir":"ltr"},{"lang":"tly","autonym":"tolışi","dir":"ltr"},{"lang":"tn","autonym":"Setswana","dir":"ltr"},{"lang":"to","autonym":"lea faka-Tonga","dir":"ltr"},{"lang":"tpi","autonym":"Tok Pisin","dir":"ltr"},{"lang":"tr", "autonym":"Türkçe","dir":"ltr"},{"lang":"trv","autonym":"Seediq","dir":"ltr"},{"lang":"ts","autonym":"Xitsonga","dir":"ltr"},{"lang":"tt","autonym":"татарча / tatarça","dir":"ltr"},{"lang":"tum","autonym":"chiTumbuka","dir":"ltr"},{"lang":"tw","autonym":"Twi","dir":"ltr"},{"lang":"ty","autonym":"reo tahiti","dir":"ltr"},{"lang":"tyv","autonym":"тыва дыл","dir":"ltr"},{"lang":"udm","autonym":"удмурт","dir":"ltr"},{"lang":"ur","autonym":"اردو","dir":"rtl"},{"lang":"uz","autonym":"oʻzbekcha / ўзбекча","dir":"ltr"},{"lang":"ve","autonym":"Tshivenda","dir":"ltr"},{"lang":"vec","autonym":"vèneto","dir":"ltr"},{"lang":"vep","autonym":"vepsän kel’","dir":"ltr"},{"lang":"vi","autonym":"Tiếng Việt","dir":"ltr"},{"lang":"vls","autonym":"West-Vlams","dir":"ltr"},{"lang":"vo","autonym":"Volapük","dir":"ltr"},{"lang":"vro","autonym":"võro","dir":"ltr"},{"lang":"wa","autonym":"walon","dir":"ltr"},{"lang":"war","autonym":"Winaray","dir":"ltr"},{"lang":"wo", "autonym":"Wolof","dir":"ltr"},{"lang":"wuu","autonym":"吴语","dir":"ltr"},{"lang":"xal","autonym":"хальмг","dir":"ltr"},{"lang":"xh","autonym":"isiXhosa","dir":"ltr"},{"lang":"xmf","autonym":"მარგალური","dir":"ltr"},{"lang":"yi","autonym":"ייִדיש","dir":"rtl"},{"lang":"yo","autonym":"Yorùbá","dir":"ltr"},{"lang":"yue","autonym":"粵語","dir":"ltr"},{"lang":"za","autonym":"Vahcuengh","dir":"ltr"},{"lang":"zgh","autonym":"ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ","dir":"ltr"},{"lang":"zh","autonym":"中文","dir":"ltr"},{"lang":"zu","autonym":"isiZulu","dir":"ltr"}],"wgSectionTranslationTargetLanguages":["ace","ady","alt","am","ami","an","ang","ann","anp","ar","ary","arz","as","ast","av","avk","awa","ay","az","azb","ba","ban","bar","bbc","bcl","bdr","be","bew","bg","bho","bi","bjn","blk","bm","bn","bo","bpy","br","bs","btm","bug","ca","cdo","ce","ceb","ch","chr","ckb","co","cr","crh","cs","cu","cy","da","dag","de","dga","din","diq","dsb","dtp" ,"dv","dz","ee","el","eml","eo","es","et","eu","fa","fat","ff","fi","fj","fo","fon","fr","frp","frr","fur","fy","gag","gan","gcr","gl","glk","gn","gom","gor","gpe","gu","guc","gur","guw","gv","ha","hak","haw","he","hi","hif","hr","hsb","ht","hu","hy","hyw","ia","iba","ie","ig","igl","ilo","io","is","it","iu","ja","jam","jv","ka","kaa","kab","kbd","kbp","kcg","kg","kge","ki","kk","kl","km","kn","ko","koi","krc","ks","ku","kus","kv","kw","ky","lad","lb","lez","lg","li","lij","lld","lmo","ln","lo","lt","ltg","lv","mad","mai","map-bms","mdf","mg","mhr","mi","min","mk","ml","mn","mni","mnw","mos","mr","mrj","ms","mt","mwl","my","myv","mzn","nah","nan","nap","nb","nds","nds-nl","ne","new","nia","nl","nn","nqo","nr","nso","ny","oc","om","or","os","pa","pag","pam","pap","pcd","pcm","pdc","pl","pms","pnb","ps","pt","pwn","qu","rm","rn","ro","rsk","rue","rup","rw","sa","sah","sat","sc","scn","sco","sd","se","sg","sgs","sh","shi","shn","si","sk","skr","sl","sm","smn","sn","so","sq","sr","srn", "ss","st","stq","su","sv","sw","szl","ta","tay","tcy","tdd","te","tet","tg","th","ti","tk","tl","tly","tn","to","tpi","tr","trv","ts","tt","tum","tw","ty","tyv","udm","ur","uz","ve","vec","vep","vi","vls","vo","vro","wa","war","wo","wuu","xal","xh","xmf","yi","yo","yue","za","zgh","zh","zu"],"isLanguageSearcherCXEntrypointEnabled":true,"mintEntrypointLanguages":["ace","ast","azb","bcl","bjn","bh","crh","ff","fon","ig","is","ki","ks","lmo","min","sat","ss","tn","vec"],"wgWikibaseItemId":"Q1338010","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false,"wgMinervaPermissions":{"watchable":true,"watch":false},"wgMinervaFeatures":{"beta":false,"donate":true,"mobileOptionsLink":true,"categories":false,"pageIssues":true,"talkAtTop":true, "historyInPageActions":false,"overflowSubmenu":false,"tabsOnSpecials":true,"personalMenu":false,"mainMenuExpanded":false,"echo":true,"nightMode":true},"wgMinervaDownloadNamespaces":[0]};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.minerva.styles":"ready","skins.minerva.content.styles.images":"ready","mediawiki.hlist":"ready","skins.minerva.codex.styles":"ready","skins.minerva.icons":"ready","skins.minerva.amc.styles":"ready","ext.wikimediamessages.styles":"ready","mobile.init.styles":"ready","ext.relatedArticles.styles":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready","skins.minerva.scripts","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.switcher","ext.urlShortener.toolbar", "ext.centralauth.centralautologin","ext.popups","mobile.init","ext.echo.centralauth","ext.relatedArticles.readMore.bootstrap","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.cx.eventlogging.campaigns","ext.cx.entrypoints.mffrequentlanguages","ext.cx.entrypoints.languagesearcher.init","mw.externalguidance.init","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.relatedArticles.styles%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cmediawiki.hlist%7Cmobile.init.styles%7Cskins.minerva.amc.styles%7Cskins.minerva.codex.styles%7Cskins.minerva.content.styles.images%7Cskins.minerva.icons%2Cstyles%7Cwikibase.client.init&only=styles&skin=minerva"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=minerva"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=minerva"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="theme-color" content="#eaecf0"> <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes, minimum-scale=0.25, maximum-scale=5.0"> <meta property="og:title" content="Notation in probability and statistics - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="manifest" href="/w/api.php?action=webapp-manifest"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Notation_in_probability_and_statistics"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Notation_in_probability_and_statistics rootpage-Notation_in_probability_and_statistics stable issues-group-B skin-minerva action-view skin--responsive mw-mf-amc-disabled mw-mf"><div id="mw-mf-viewport"> <div id="mw-mf-page-center"> <a class="mw-mf-page-center__mask" href="#"></a> <header class="header-container header-chrome"> <div class="minerva-header"> <nav class="navigation-drawer toggle-list view-border-box"> <input type="checkbox" id="main-menu-input" class="toggle-list__checkbox" role="button" aria-haspopup="true" aria-expanded="false" aria-labelledby="mw-mf-main-menu-button"> <label role="button" for="main-menu-input" id="mw-mf-main-menu-button" aria-hidden="true" data-event-name="ui.mainmenu" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet toggle-list__toggle"> <span class="minerva-icon minerva-icon--menu"></span> <span></span> </label> <div id="mw-mf-page-left" class="menu view-border-box"> <ul id="p-navigation" class="toggle-list__list"> <li class="toggle-list-item "> <a class="toggle-list-item__anchor menu__item--home" href="/wiki/Main_Page" data-mw="interface"> <span class="minerva-icon minerva-icon--home"></span> <span class="toggle-list-item__label">Home</span> </a> </li> <li class="toggle-list-item "> <a class="toggle-list-item__anchor menu__item--random" href="/wiki/Special:Random" data-mw="interface"> <span class="minerva-icon minerva-icon--die"></span> <span class="toggle-list-item__label">Random</span> </a> </li> <li class="toggle-list-item skin-minerva-list-item-jsonly"> <a class="toggle-list-item__anchor menu__item--nearby" href="/wiki/Special:Nearby" data-event-name="menu.nearby" data-mw="interface"> <span class="minerva-icon minerva-icon--mapPin"></span> <span class="toggle-list-item__label">Nearby</span> </a> </li> </ul> <ul id="p-personal" class="toggle-list__list"> <li class="toggle-list-item "> <a class="toggle-list-item__anchor menu__item--login" href="/w/index.php?title=Special:UserLogin&returnto=Notation+in+probability+and+statistics" data-event-name="menu.login" data-mw="interface"> <span class="minerva-icon minerva-icon--logIn"></span> <span class="toggle-list-item__label">Log in</span> </a> </li> </ul> <ul id="pt-preferences" class="toggle-list__list"> <li class="toggle-list-item skin-minerva-list-item-jsonly"> <a class="toggle-list-item__anchor menu__item--settings" href="/w/index.php?title=Special:MobileOptions&returnto=Notation+in+probability+and+statistics" data-event-name="menu.settings" data-mw="interface"> <span class="minerva-icon minerva-icon--settings"></span> <span class="toggle-list-item__label">Settings</span> </a> </li> </ul> <ul id="p-donation" class="toggle-list__list"> <li class="toggle-list-item "> <a class="toggle-list-item__anchor menu__item--donate" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en&utm_key=minerva" data-event-name="menu.donate" data-mw="interface"> <span class="minerva-icon minerva-icon--heart"></span> <span class="toggle-list-item__label">Donate</span> </a> </li> </ul> <ul class="hlist"> <li class="toggle-list-item "> <a class="toggle-list-item__anchor menu__item--about" href="/wiki/Wikipedia:About" data-mw="interface"> <span class="toggle-list-item__label">About Wikipedia</span> </a> </li> <li class="toggle-list-item "> <a class="toggle-list-item__anchor menu__item--disclaimers" href="/wiki/Wikipedia:General_disclaimer" data-mw="interface"> <span class="toggle-list-item__label">Disclaimers</span> </a> </li> </ul> </div> <label class="main-menu-mask" for="main-menu-input"></label> </nav> <div class="branding-box"> <a href="/wiki/Main_Page"> <span><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </span> </a> </div> <form action="/w/index.php" method="get" class="minerva-search-form"> <div class="search-box"> <input type="hidden" name="title" value="Special:Search"/> <input class="search skin-minerva-search-trigger" id="searchInput" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f"> <span class="search-box-icon-overlay"><span class="minerva-icon minerva-icon--search"></span> </span> </div> <button id="searchIcon" class="cdx-button cdx-button--size-large cdx-button--icon-only cdx-button--weight-quiet skin-minerva-search-trigger"> <span class="minerva-icon minerva-icon--search"></span> <span>Search</span> </button> </form> <nav class="minerva-user-navigation" aria-label="User navigation"> </nav> </div> </header> <main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Notation in probability and statistics</span></h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Notation_in_probability_and_statistics" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Notation_in_probability_and_statistics" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> <span class="minerva-icon minerva-icon--language"></span> <span>Language</span> </a> </li> <li id="page-actions-watch" class="page-actions-menu__list-item"> <a role="button" id="ca-watch" href="/w/index.php?title=Special:UserLogin&returnto=Notation+in+probability+and+statistics" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> <span class="minerva-icon minerva-icon--star"></span> <span>Watch</span> </a> </li> <li id="page-actions-edit" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"><p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"> <p><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a> and <a href="/wiki/Statistics" title="Statistics">statistics</a> have some commonly used conventions, in addition to standard <a href="/wiki/Mathematical_notation" title="Mathematical notation">mathematical notation</a> and <a href="/wiki/Table_of_mathematical_symbols" class="mw-redirect" title="Table of mathematical symbols">mathematical symbols</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Probability_theory"><span class="tocnumber">1</span> <span class="toctext">Probability theory</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Statistics"><span class="tocnumber">2</span> <span class="toctext">Statistics</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Critical_values"><span class="tocnumber">3</span> <span class="toctext">Critical values</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Linear_algebra"><span class="tocnumber">4</span> <span class="toctext">Linear algebra</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Abbreviations"><span class="tocnumber">5</span> <span class="toctext">Abbreviations</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#External_links"><span class="tocnumber">8</span> <span class="toctext">External links</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Probability_theory">Probability theory</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=1" title="Edit section: Probability theory" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Notation_in_probability_and_statistics" title="Special:EditPage/Notation in probability and statistics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <ul><li><a href="/wiki/Random_variable" title="Random variable">Random variables</a> are usually written in <a href="/wiki/Upper_case" class="mw-redirect" title="Upper case">upper case</a> Roman letters, such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\textstyle X}"></noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" data-alt="{\textstyle X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle Y}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6222becc4f0c5effa012e5335b170575fdbbaad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\textstyle Y}"></noscript><span class="lazy-image-placeholder" style="width: 1.773ex;height: 2.009ex;vertical-align: -0.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6222becc4f0c5effa012e5335b170575fdbbaad3" data-alt="{\textstyle Y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable. They do not represent a single number or a single category. For instance, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X=x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X=x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/073aa90e978ecf7072bb4afcf04ba2ea00140803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.963ex; height:2.843ex;" alt="{\displaystyle P(X=x)}"></noscript><span class="lazy-image-placeholder" style="width: 9.963ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/073aa90e978ecf7072bb4afcf04ba2ea00140803" data-alt="{\displaystyle P(X=x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is written, then it represents the probability that a particular realisation of a random variable (e.g., height, number of cars, or bicycle colour), <i>X</i>, would be equal to a particular value or category (e.g., 1.735 m, 52, or purple), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" data-alt="{\textstyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. It is important that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\textstyle X}"></noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" data-alt="{\textstyle X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" data-alt="{\textstyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are not confused into meaning the same thing. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\textstyle X}"></noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" data-alt="{\textstyle X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is an idea, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" data-alt="{\textstyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is a value. Clearly they are related, but they do not have identical meanings.</li> <li>Particular realisations of a random variable are written in corresponding <a href="/wiki/Lower_case" class="mw-redirect" title="Lower case">lower case</a> letters. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1},x_{2},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x_{1},x_{2},\ldots ,x_{n}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44d30d0c6dcc5a992750d7dfe6448fd326529e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\textstyle x_{1},x_{2},\ldots ,x_{n}}"></noscript><span class="lazy-image-placeholder" style="width: 13.528ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44d30d0c6dcc5a992750d7dfe6448fd326529e3" data-alt="{\textstyle x_{1},x_{2},\ldots ,x_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> could be a <a href="/wiki/Random_sample" class="mw-redirect" title="Random sample">sample</a> corresponding to the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\textstyle X}"></noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" data-alt="{\textstyle X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. A cumulative probability is formally written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X\leq x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X\leq x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eb4eb6132dced98973b8293d25c1676c7ebecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.963ex; height:2.843ex;" alt="{\displaystyle P(X\leq x)}"></noscript><span class="lazy-image-placeholder" style="width: 9.963ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0eb4eb6132dced98973b8293d25c1676c7ebecf" data-alt="{\displaystyle P(X\leq x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to distinguish the random variable from its realization.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>The probability is sometimes written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }"></noscript><span class="lazy-image-placeholder" style="width: 1.42ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" data-alt="{\displaystyle \mathbb {P} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to distinguish it from other functions and measure <i>P</i> to avoid having to define "<i>P</i> is a probability" and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (X\in A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>∈</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (X\in A)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3df8d77bf466ebe9a7943c85f5b4bb8816ad81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.793ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (X\in A)}"></noscript><span class="lazy-image-placeholder" style="width: 9.793ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3df8d77bf466ebe9a7943c85f5b4bb8816ad81" data-alt="{\displaystyle \mathbb {P} (X\in A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is short for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>A</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df6935102515fb62f59cfbe654ad127a639cfda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.6ex; height:2.843ex;" alt="{\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}"></noscript><span class="lazy-image-placeholder" style="width: 23.6ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df6935102515fb62f59cfbe654ad127a639cfda" data-alt="{\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" data-alt="{\displaystyle \Omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the event space, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></noscript><span class="lazy-image-placeholder" style="width: 1.98ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" data-alt="{\displaystyle X}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is a random variable that is a function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></noscript><span class="lazy-image-placeholder" style="width: 1.446ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" data-alt="{\displaystyle \omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (i.e., it depends upon <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></noscript><span class="lazy-image-placeholder" style="width: 1.446ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" data-alt="{\displaystyle \omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></noscript><span class="lazy-image-placeholder" style="width: 1.446ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" data-alt="{\displaystyle \omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is some outcome of interest within the domain specified by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" data-alt="{\displaystyle \Omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (say, a particular height, or a particular colour of a car). <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(A)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73acce4b7989bbfd543e36d2be5f6e433a9bbaf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.047ex; height:2.843ex;" alt="{\displaystyle \Pr(A)}"></noscript><span class="lazy-image-placeholder" style="width: 6.047ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73acce4b7989bbfd543e36d2be5f6e433a9bbaf4" data-alt="{\displaystyle \Pr(A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> notation is used alternatively.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (A\cap B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (A\cap B)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c438f98015447ef3671f2280c7825cf6f12a6379" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.319ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (A\cap B)}"></noscript><span class="lazy-image-placeholder" style="width: 9.319ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c438f98015447ef3671f2280c7825cf6f12a6379" data-alt="{\displaystyle \mathbb {P} (A\cap B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} [B\cap A]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">[</mo> <mi>B</mi> <mo>∩</mo> <mi>A</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} [B\cap A]}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34244c1cb5954da1d33e683d5fa89e4e7df30959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.804ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} [B\cap A]}"></noscript><span class="lazy-image-placeholder" style="width: 8.804ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34244c1cb5954da1d33e683d5fa89e4e7df30959" data-alt="{\displaystyle \mathbb {P} [B\cap A]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> indicates the probability that events <i>A</i> and <i>B</i> both occur. The <a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> of random variables <i>X</i> and <i>Y</i> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X,Y)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fd347c3c83e69250fe5a6c3648fed056ff8f68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.342ex; height:2.843ex;" alt="{\displaystyle P(X,Y)}"></noscript><span class="lazy-image-placeholder" style="width: 8.342ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fd347c3c83e69250fe5a6c3648fed056ff8f68" data-alt="{\displaystyle P(X,Y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, while joint probability mass function or probability density function as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29473ed0c4e838ac9dbe074535e507166c0e9101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}"></noscript><span class="lazy-image-placeholder" style="width: 6.607ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29473ed0c4e838ac9dbe074535e507166c0e9101" data-alt="{\displaystyle f(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and joint cumulative distribution function as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81df93097aee89e43fd71a8f4f744c418e1e2d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.069ex; height:2.843ex;" alt="{\displaystyle F(x,y)}"></noscript><span class="lazy-image-placeholder" style="width: 7.069ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81df93097aee89e43fd71a8f4f744c418e1e2d4d" data-alt="{\displaystyle F(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (A\cup B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (A\cup B)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37a66b293d89bb2e2a92975afb13856bc477ec55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.319ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (A\cup B)}"></noscript><span class="lazy-image-placeholder" style="width: 9.319ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37a66b293d89bb2e2a92975afb13856bc477ec55" data-alt="{\displaystyle \mathbb {P} (A\cup B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} [B\cup A]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">[</mo> <mi>B</mi> <mo>∪</mo> <mi>A</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} [B\cup A]}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f6eec48e283636dc59510301e8024854d1f57fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.804ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} [B\cup A]}"></noscript><span class="lazy-image-placeholder" style="width: 8.804ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f6eec48e283636dc59510301e8024854d1f57fa" data-alt="{\displaystyle \mathbb {P} [B\cup A]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> indicates the probability of either event <i>A</i> or event <i>B</i> occurring ("or" in this case means <a href="/wiki/Inclusive_or" class="mw-redirect" title="Inclusive or">one or the other or both</a>).</li> <li><a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">σ-algebras</a> are usually written with uppercase <a href="/wiki/Calligraphy" title="Calligraphy">calligraphic</a> (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.927ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" data-alt="{\displaystyle {\mathcal {F}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the set of sets on which we define the probability <i>P</i>)</li> <li><a href="/wiki/Probability_density_function" title="Probability density function">Probability density functions</a> (pdfs) and <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass functions</a> are denoted by lowercase letters, e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></noscript><span class="lazy-image-placeholder" style="width: 4.418ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" data-alt="{\displaystyle f(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{X}(x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29310a010e7f7dfa33ba69bcf1ef9ec166d461dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.911ex; height:2.843ex;" alt="{\displaystyle f_{X}(x)}"></noscript><span class="lazy-image-placeholder" style="width: 5.911ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29310a010e7f7dfa33ba69bcf1ef9ec166d461dd" data-alt="{\displaystyle f_{X}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">Cumulative distribution functions</a> (cdfs) are denoted by uppercase letters, e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></noscript><span class="lazy-image-placeholder" style="width: 4.88ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" data-alt="{\displaystyle F(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{X}(x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242727215e028fc47529c5bd7035e88cc0da25e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.266ex; height:2.843ex;" alt="{\displaystyle F_{X}(x)}"></noscript><span class="lazy-image-placeholder" style="width: 6.266ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242727215e028fc47529c5bd7035e88cc0da25e0" data-alt="{\displaystyle F_{X}(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li><a href="/wiki/Survival_function" title="Survival function">Survival functions</a> or complementary cumulative distribution functions are often denoted by placing an <a href="/wiki/Overbar" class="mw-redirect" title="Overbar">overbar</a> over the symbol for the cumulative:<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {F}}(x)=1-F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {F}}(x)=1-F(x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1941dc5171830203a01848eab8b8fba7fe36503c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.136ex; height:3.509ex;" alt="{\displaystyle {\overline {F}}(x)=1-F(x)}"></noscript><span class="lazy-image-placeholder" style="width: 17.136ex;height: 3.509ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1941dc5171830203a01848eab8b8fba7fe36503c" data-alt="{\displaystyle {\overline {F}}(x)=1-F(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, or denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(x)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1886b5a535ed8f168a7c3a83afc8ca440edcdc6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle S(x)}"></noscript><span class="lazy-image-placeholder" style="width: 4.638ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1886b5a535ed8f168a7c3a83afc8ca440edcdc6d" data-alt="{\displaystyle S(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</li> <li>In particular, the pdf of the <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a> is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varphi (z)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1358bdf1286826fffd8e70843233399b5d69f8ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\textstyle \varphi (z)}"></noscript><span class="lazy-image-placeholder" style="width: 4.418ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1358bdf1286826fffd8e70843233399b5d69f8ee" data-alt="{\textstyle \varphi (z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, and its cdf by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi (z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi (z)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dad4994c124394affbcf0d08c3ac94d238b12eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\textstyle \Phi (z)}"></noscript><span class="lazy-image-placeholder" style="width: 4.576ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dad4994c124394affbcf0d08c3ac94d238b12eb" data-alt="{\textstyle \Phi (z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li>Some common operators:</li></ul> <dl><dd><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {E} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathrm {E} [X]}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc34eba3b7c1f28e7ff3ce1c16a279b20f8c459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.857ex; height:2.843ex;" alt="{\textstyle \mathrm {E} [X]}"></noscript><span class="lazy-image-placeholder" style="width: 4.857ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc34eba3b7c1f28e7ff3ce1c16a279b20f8c459" data-alt="{\textstyle \mathrm {E} [X]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> : <a href="/wiki/Expected_value" title="Expected value">expected value</a> of <i>X</i></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {var} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {var} [X]}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58e4e8fdec756ccef1f1616d51a192c4e460e960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.575ex; height:2.843ex;" alt="{\textstyle \operatorname {var} [X]}"></noscript><span class="lazy-image-placeholder" style="width: 6.575ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58e4e8fdec756ccef1f1616d51a192c4e460e960" data-alt="{\textstyle \operatorname {var} [X]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> : <a href="/wiki/Variance" title="Variance">variance</a> of <i>X</i></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {cov} [X,Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {cov} [X,Y]}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca33c39cecc1eb846c11b770117609acb8c08131" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.503ex; height:2.843ex;" alt="{\textstyle \operatorname {cov} [X,Y]}"></noscript><span class="lazy-image-placeholder" style="width: 9.503ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca33c39cecc1eb846c11b770117609acb8c08131" data-alt="{\textstyle \operatorname {cov} [X,Y]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> : <a href="/wiki/Covariance" title="Covariance">covariance</a> of <i>X</i> and <i>Y</i></li></ul></dd></dl> <ul><li>X is independent of Y is often written <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\perp Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊥</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\perp Y}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0214f32baeaae3f2ac4b70f2a0f05a30217ad1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle X\perp Y}"></noscript><span class="lazy-image-placeholder" style="width: 6.852ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0214f32baeaae3f2ac4b70f2a0f05a30217ad1b" data-alt="{\displaystyle X\perp Y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\perp \!\!\!\perp Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊥</mo> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mo>⊥</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\perp \!\!\!\perp Y}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952932989c58b2287a97b193bb5d62449fddbb24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.499ex; height:2.176ex;" alt="{\displaystyle X\perp \!\!\!\perp Y}"></noscript><span class="lazy-image-placeholder" style="width: 7.499ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/952932989c58b2287a97b193bb5d62449fddbb24" data-alt="{\displaystyle X\perp \!\!\!\perp Y}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, and X is independent of Y given W is often written</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\perp \!\!\!\perp Y\,|\,W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊥</mo> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mo>⊥</mo> <mi>Y</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\perp \!\!\!\perp Y\,|\,W}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdd6fff8776a930ccaba091224bb1958e6360c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.355ex; height:2.843ex;" alt="{\displaystyle X\perp \!\!\!\perp Y\,|\,W}"></noscript><span class="lazy-image-placeholder" style="width: 11.355ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdd6fff8776a930ccaba091224bb1958e6360c8" data-alt="{\displaystyle X\perp \!\!\!\perp Y\,|\,W}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\perp Y\,|\,W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>⊥</mo> <mi>Y</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace"></mspace> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\perp Y\,|\,W}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae6d018f1080debde6565000e3ba82ff7c7ca853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.708ex; height:2.843ex;" alt="{\displaystyle X\perp Y\,|\,W}"></noscript><span class="lazy-image-placeholder" style="width: 10.708ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae6d018f1080debde6565000e3ba82ff7c7ca853" data-alt="{\displaystyle X\perp Y\,|\,W}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle P(A\mid B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∣</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle P(A\mid B)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0845271c02c44c9918c1464f4393ddc1cc524499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.999ex; height:2.843ex;" alt="{\displaystyle \textstyle P(A\mid B)}"></noscript><span class="lazy-image-placeholder" style="width: 8.999ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0845271c02c44c9918c1464f4393ddc1cc524499" data-alt="{\displaystyle \textstyle P(A\mid B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, the <i><a href="/wiki/Conditional_probability" title="Conditional probability">conditional probability</a></i>, is the probability of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>A</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle A}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/064bd4054ef41bef35dbad11c97c9bd46015c0ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \textstyle A}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/064bd4054ef41bef35dbad11c97c9bd46015c0ab" data-alt="{\displaystyle \textstyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <i>given</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>B</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle B}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d11529def8e8af1c97b2d5fc86f4cad5cb0a4c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle \textstyle B}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d11529def8e8af1c97b2d5fc86f4cad5cb0a4c2" data-alt="{\displaystyle \textstyle B}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Statistics">Statistics</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=2" title="Edit section: Statistics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Notation_in_probability_and_statistics" title="Special:EditPage/Notation in probability and statistics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <ul><li>Greek letters (e.g. <i>θ</i>, <i>β</i>) are commonly used to denote unknown parameters (population parameters).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>A tilde (~) denotes "has the probability distribution of".</li> <li>Placing a hat, or caret (also known as a circumflex), over a true parameter denotes an <a href="/wiki/Estimator" title="Estimator">estimator</a> of it, e.g., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\theta }}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a031ed08d81ed4f0df984f0d9f30b7ae07be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.356ex; height:2.843ex;" alt="{\displaystyle {\widehat {\theta }}}"></noscript><span class="lazy-image-placeholder" style="width: 1.356ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a031ed08d81ed4f0df984f0d9f30b7ae07be46" data-alt="{\displaystyle {\widehat {\theta }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is an estimator for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></noscript><span class="lazy-image-placeholder" style="width: 1.09ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" data-alt="{\displaystyle \theta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li>The <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> of a series of values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{1},x_{2},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x_{1},x_{2},\ldots ,x_{n}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44d30d0c6dcc5a992750d7dfe6448fd326529e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\textstyle x_{1},x_{2},\ldots ,x_{n}}"></noscript><span class="lazy-image-placeholder" style="width: 13.528ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44d30d0c6dcc5a992750d7dfe6448fd326529e3" data-alt="{\textstyle x_{1},x_{2},\ldots ,x_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is often denoted by placing an "<a href="/wiki/Overbar" class="mw-redirect" title="Overbar">overbar</a>" over the symbol, e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" data-alt="{\displaystyle {\bar {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, pronounced "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" data-alt="{\textstyle x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> bar".</li> <li>Some commonly used symbols for <a href="/wiki/Sample_(statistics)" class="mw-redirect" title="Sample (statistics)">sample</a> statistics are given below: <ul><li>the <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\displaystyle {\bar {x}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" data-alt="{\displaystyle {\bar {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</li> <li>the <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s^{2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ac5be3cbea7265af53b722b53ab3077320e713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.509ex;" alt="{\textstyle s^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.145ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ac5be3cbea7265af53b722b53ab3077320e713" data-alt="{\textstyle s^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</li> <li>the <a href="/wiki/Sample_standard_deviation" class="mw-redirect" title="Sample standard deviation">sample standard deviation</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle s}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45013da7f502d373d039cb1056a9c4d1ea06ffc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\textstyle s}"></noscript><span class="lazy-image-placeholder" style="width: 1.09ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45013da7f502d373d039cb1056a9c4d1ea06ffc7" data-alt="{\textstyle s}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i>,</li> <li>the <a href="/wiki/Pearson_correlation_coefficient" title="Pearson correlation coefficient">sample correlation coefficient</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\textstyle r}"></noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729" data-alt="{\textstyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i>,</li> <li>the sample cumulants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k_{r}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b197b962f1f51d76e03fcab39a98918b662e936a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.185ex; height:2.509ex;" alt="{\textstyle k_{r}}"></noscript><span class="lazy-image-placeholder" style="width: 2.185ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b197b962f1f51d76e03fcab39a98918b662e936a" data-alt="{\textstyle k_{r}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li></ul></li> <li>Some commonly used symbols for <a href="/wiki/Statistical_population" title="Statistical population">population</a> parameters are given below: <ul><li>the population mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mu }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/259577540a13444806174d5a1ae7662974f58085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\textstyle \mu }"></noscript><span class="lazy-image-placeholder" style="width: 1.402ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/259577540a13444806174d5a1ae7662974f58085" data-alt="{\textstyle \mu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</li> <li>the population variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma ^{2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86f1d00f664920ef46109bcddc0778f4976b490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.509ex;" alt="{\textstyle \sigma ^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.385ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a86f1d00f664920ef46109bcddc0778f4976b490" data-alt="{\textstyle \sigma ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>,</li> <li>the population standard deviation <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sigma }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca336f17488923d11206e131f7eb2a569e8dde2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle \sigma }"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca336f17488923d11206e131f7eb2a569e8dde2" data-alt="{\textstyle \sigma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i>,</li> <li>the population <a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">correlation</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \rho }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe88055cce2aa2861ff06d1386e021822fad300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\textstyle \rho }"></noscript><span class="lazy-image-placeholder" style="width: 1.202ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe88055cce2aa2861ff06d1386e021822fad300" data-alt="{\textstyle \rho }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i>,</li> <li>the population <a href="/wiki/Cumulant" title="Cumulant">cumulants</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \kappa _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \kappa _{r}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4b3e03c1f4c1bb1eebd66c955278c46de46b40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.313ex; height:2.009ex;" alt="{\textstyle \kappa _{r}}"></noscript><span class="lazy-image-placeholder" style="width: 2.313ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4b3e03c1f4c1bb1eebd66c955278c46de46b40" data-alt="{\textstyle \kappa _{r}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></i>,</li></ul></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{(k)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{(k)}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3387edfb141c35ca899ddd548aeeb7f34c1781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.698ex; height:2.509ex;" alt="{\displaystyle x_{(k)}}"></noscript><span class="lazy-image-placeholder" style="width: 3.698ex;height: 2.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3387edfb141c35ca899ddd548aeeb7f34c1781" data-alt="{\displaystyle x_{(k)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is used for the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{\text{th}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{\text{th}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4972c293a203ef44247bcdc01bc77d6df0e21719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.997ex; height:2.676ex;" alt="{\displaystyle k^{\text{th}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.997ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4972c293a203ef44247bcdc01bc77d6df0e21719" data-alt="{\displaystyle k^{\text{th}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <a href="/wiki/Order_statistic" title="Order statistic">order statistic</a>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{(1)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{(1)}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/464c1aa643240b7e29d9a21be0f4c1c75d1c20d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.663ex; height:2.509ex;" alt="{\displaystyle x_{(1)}}"></noscript><span class="lazy-image-placeholder" style="width: 3.663ex;height: 2.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/464c1aa643240b7e29d9a21be0f4c1c75d1c20d4" data-alt="{\displaystyle x_{(1)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the sample minimum and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{(n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{(n)}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6ad08f22682b82919b668b79cde43017aa102e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.828ex; height:2.509ex;" alt="{\displaystyle x_{(n)}}"></noscript><span class="lazy-image-placeholder" style="width: 3.828ex;height: 2.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6ad08f22682b82919b668b79cde43017aa102e" data-alt="{\displaystyle x_{(n)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the sample maximum from a total sample size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" data-alt="{\textstyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Critical_values">Critical values</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=3" title="Edit section: Critical values" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Notation_in_probability_and_statistics" title="Special:EditPage/Notation in probability and statistics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The <i>α</i>-level upper <a href="/wiki/Critical_value_(statistics)" class="mw-redirect" title="Critical value (statistics)">critical value</a> of a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> is the value exceeded with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d86dbd6183264b2f8569da1751380b173c7b185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\textstyle \alpha }"></noscript><span class="lazy-image-placeholder" style="width: 1.488ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d86dbd6183264b2f8569da1751380b173c7b185" data-alt="{\textstyle \alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, that is, the value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x_{\alpha }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a087773da36998fd9645e8c09fa783963c9c0cc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.614ex; height:2.009ex;" alt="{\textstyle x_{\alpha }}"></noscript><span class="lazy-image-placeholder" style="width: 2.614ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a087773da36998fd9645e8c09fa783963c9c0cc4" data-alt="{\textstyle x_{\alpha }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F(x_{\alpha })=1-\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F(x_{\alpha })=1-\alpha }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f989e7cc34e9f8625727f9ff4dda983387b7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.753ex; height:2.843ex;" alt="{\textstyle F(x_{\alpha })=1-\alpha }"></noscript><span class="lazy-image-placeholder" style="width: 14.753ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f989e7cc34e9f8625727f9ff4dda983387b7d8" data-alt="{\textstyle F(x_{\alpha })=1-\alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8256dae10b9e3abb3592ff608e81c8bc324edce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\textstyle F}"></noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8256dae10b9e3abb3592ff608e81c8bc324edce3" data-alt="{\textstyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z_{\alpha }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbbd19c2c92697f49fb0395f87b2ae3462c3148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.365ex; height:2.009ex;" alt="{\textstyle z_{\alpha }}"></noscript><span class="lazy-image-placeholder" style="width: 2.365ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbbd19c2c92697f49fb0395f87b2ae3462c3148" data-alt="{\textstyle z_{\alpha }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle z(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle z(\alpha )}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a534b8b2fc3ab2a6d83f73e52a4e87142108b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.385ex; height:2.843ex;" alt="{\textstyle z(\alpha )}"></noscript><span class="lazy-image-placeholder" style="width: 4.385ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a534b8b2fc3ab2a6d83f73e52a4e87142108b20" data-alt="{\textstyle z(\alpha )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the <a href="/wiki/Standard_normal_distribution" class="mw-redirect" title="Standard normal distribution">standard normal distribution</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t_{\alpha ,\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t_{\alpha ,\nu }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d55129b0d9c64560430a49e326e9da8b9921892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.452ex; height:2.676ex;" alt="{\textstyle t_{\alpha ,\nu }}"></noscript><span class="lazy-image-placeholder" style="width: 3.452ex;height: 2.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d55129b0d9c64560430a49e326e9da8b9921892" data-alt="{\textstyle t_{\alpha ,\nu }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t(\alpha ,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t(\alpha ,\nu )}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e7a3ebf5fd6320f6a63ea4fdba165bf714a263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.403ex; height:2.843ex;" alt="{\textstyle t(\alpha ,\nu )}"></noscript><span class="lazy-image-placeholder" style="width: 6.403ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65e7a3ebf5fd6320f6a63ea4fdba165bf714a263" data-alt="{\textstyle t(\alpha ,\nu )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the <a href="/wiki/Student%27s_t-distribution" title="Student's t-distribution"><i>t</i>-distribution</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \nu }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9518d669addb0cc614b524a4a5753a69b6f1b9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\textstyle \nu }"></noscript><span class="lazy-image-placeholder" style="width: 1.232ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9518d669addb0cc614b524a4a5753a69b6f1b9e3" data-alt="{\textstyle \nu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degrees of freedom</a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\chi _{\alpha ,\nu }}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <mi>ν</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\chi _{\alpha ,\nu }}^{2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b2efe49df11e316fc39e00e078129c959bc405" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.122ex; height:3.343ex;" alt="{\displaystyle {\chi _{\alpha ,\nu }}^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 5.122ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b2efe49df11e316fc39e00e078129c959bc405" data-alt="{\displaystyle {\chi _{\alpha ,\nu }}^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\chi }^{2}(\alpha ,\nu )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>χ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\chi }^{2}(\alpha ,\nu )}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e586ef05dbf25e039be0fa88f53fc6eb0829bf65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.072ex; height:3.176ex;" alt="{\displaystyle {\chi }^{2}(\alpha ,\nu )}"></noscript><span class="lazy-image-placeholder" style="width: 8.072ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e586ef05dbf25e039be0fa88f53fc6eb0829bf65" data-alt="{\displaystyle {\chi }^{2}(\alpha ,\nu )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \nu }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9518d669addb0cc614b524a4a5753a69b6f1b9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\textstyle \nu }"></noscript><span class="lazy-image-placeholder" style="width: 1.232ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9518d669addb0cc614b524a4a5753a69b6f1b9e3" data-alt="{\textstyle \nu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> degrees of freedom</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec774db5e3d38c92d70032714c4f2ff3bd83709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.981ex; height:2.843ex;" alt="{\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}}"></noscript><span class="lazy-image-placeholder" style="width: 6.981ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec774db5e3d38c92d70032714c4f2ff3bd83709" data-alt="{\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F(\alpha ,\nu _{1},\nu _{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F(\alpha ,\nu _{1},\nu _{2})}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1185c3dcc932349b87c4ba30f8f2365827ba28f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.511ex; height:2.843ex;" alt="{\textstyle F(\alpha ,\nu _{1},\nu _{2})}"></noscript><span class="lazy-image-placeholder" style="width: 11.511ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1185c3dcc932349b87c4ba30f8f2365827ba28f6" data-alt="{\textstyle F(\alpha ,\nu _{1},\nu _{2})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> for the <a href="/wiki/F-distribution" title="F-distribution">F-distribution</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \nu _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \nu _{1}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c659b2b6e6a98eae9d9eab040a34a3b540c673" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.203ex; height:2.009ex;" alt="{\textstyle \nu _{1}}"></noscript><span class="lazy-image-placeholder" style="width: 2.203ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c659b2b6e6a98eae9d9eab040a34a3b540c673" data-alt="{\textstyle \nu _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \nu _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \nu _{2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e918f9cf8ef16adcdeb3a1a88ba31b600e6d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.203ex; height:2.009ex;" alt="{\textstyle \nu _{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.203ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e918f9cf8ef16adcdeb3a1a88ba31b600e6d09" data-alt="{\textstyle \nu _{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> degrees of freedom</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Linear_algebra">Linear algebra</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=4" title="Edit section: Linear algebra" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Notation_in_probability_and_statistics" title="Special:EditPage/Notation in probability and statistics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <ul><li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a> are usually denoted by boldface capital letters, e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathbf {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathbf {A}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427d78773ed82127761a1a0585aa3f6cc6e2399b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\textstyle {\mathbf {A}}}"></noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427d78773ed82127761a1a0585aa3f6cc6e2399b" data-alt="{\textstyle {\mathbf {A}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li><a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">Column vectors</a> are usually denoted by boldface lowercase letters, e.g. <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathbf {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathbf {x}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175f1f7543337aabb238371921b60c2186a1df0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\textstyle {\mathbf {x}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175f1f7543337aabb238371921b60c2186a1df0d" data-alt="{\textstyle {\mathbf {x}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></b>.</li> <li>The <a href="/wiki/Transpose" title="Transpose">transpose</a> operator is denoted by either a superscript T (e.g. <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathbf {A}}^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathbf {A}}^{\mathrm {T} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974a4104699d6c81800094dd150b00e22b41a9e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.438ex; height:2.676ex;" alt="{\textstyle {\mathbf {A}}^{\mathrm {T} }}"></noscript><span class="lazy-image-placeholder" style="width: 3.438ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974a4104699d6c81800094dd150b00e22b41a9e8" data-alt="{\textstyle {\mathbf {A}}^{\mathrm {T} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></b>) or a <a href="/wiki/Prime_(symbol)" title="Prime (symbol)">prime symbol</a> (e.g. <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathbf {A}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathbf {A}}'}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ec837151f86430d4850678c68c00fc9a066c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.704ex; height:2.509ex;" alt="{\textstyle {\mathbf {A}}'}"></noscript><span class="lazy-image-placeholder" style="width: 2.704ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ec837151f86430d4850678c68c00fc9a066c65" data-alt="{\textstyle {\mathbf {A}}'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></b>).</li> <li>A <a href="/wiki/Row_vector" class="mw-redirect" title="Row vector">row vector</a> is written as the transpose of a column vector, e.g. <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathbf {x}}^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathbf {x}}^{\mathrm {T} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee843cc669383a9843be86c12882ae723a32b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.83ex; height:2.509ex;" alt="{\textstyle {\mathbf {x}}^{\mathrm {T} }}"></noscript><span class="lazy-image-placeholder" style="width: 2.83ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee843cc669383a9843be86c12882ae723a32b8e" data-alt="{\textstyle {\mathbf {x}}^{\mathrm {T} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></b> or <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathbf {x}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\mathbf {x}}'}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24914322132dec958288c516ac54e221b7ad8d16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.096ex; height:2.343ex;" alt="{\textstyle {\mathbf {x}}'}"></noscript><span class="lazy-image-placeholder" style="width: 2.096ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24914322132dec958288c516ac54e221b7ad8d16" data-alt="{\textstyle {\mathbf {x}}'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></b>.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Abbreviations">Abbreviations</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=5" title="Edit section: Abbreviations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Notation_in_probability_and_statistics" title="Special:EditPage/Notation in probability and statistics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">March 2021</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Common abbreviations include: </p> <ul><li><b>a.e.</b> <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a></li> <li><b>a.s.</b> <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a></li> <li><b>cdf</b> <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a></li> <li><b>cmf</b> <a href="/wiki/Cumulative_mass_function" class="mw-redirect" title="Cumulative mass function">cumulative mass function</a></li> <li><b>df</b> <a href="/wiki/Degrees_of_freedom_(statistics)" title="Degrees of freedom (statistics)">degrees of freedom</a> (also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></noscript><span class="lazy-image-placeholder" style="width: 1.232ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" data-alt="{\displaystyle \nu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>)</li> <li><b>i.i.d.</b> <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a></li> <li><b>pdf</b> <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a></li> <li><b>pmf</b> <a href="/wiki/Probability_mass_function" title="Probability mass function">probability mass function</a></li> <li><b>r.v.</b> <a href="/wiki/Random_variable" title="Random variable">random variable</a></li> <li><b>w.p.</b> with probability; <b>wp1</b> <a href="/wiki/With_probability_1" class="mw-redirect" title="With probability 1">with probability 1</a></li> <li><b>i.o.</b> infinitely often, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A_{n}{\text{ i.o.}}\}=\bigcap _{N}\bigcup _{n\geq N}A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> i.o.</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munder> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mi>N</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A_{n}{\text{ i.o.}}\}=\bigcap _{N}\bigcup _{n\geq N}A_{n}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a91bb750ea71c16a3f83c0e677e0d3ebe12f1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.11ex; height:5.843ex;" alt="{\displaystyle \{A_{n}{\text{ i.o.}}\}=\bigcap _{N}\bigcup _{n\geq N}A_{n}}"></noscript><span class="lazy-image-placeholder" style="width: 22.11ex;height: 5.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a91bb750ea71c16a3f83c0e677e0d3ebe12f1c4" data-alt="{\displaystyle \{A_{n}{\text{ i.o.}}\}=\bigcap _{N}\bigcup _{n\geq N}A_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></li> <li><b>ult.</b> ultimately, i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> ult.</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munder> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mi>N</mi> </mrow> </munder> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e894e0924c171d14559cc532ffa0aa43538f9a8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.498ex; height:5.843ex;" alt="{\displaystyle \{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}}"></noscript><span class="lazy-image-placeholder" style="width: 22.498ex;height: 5.843ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e894e0924c171d14559cc532ffa0aa43538f9a8d" data-alt="{\displaystyle \{A_{n}{\text{ ult.}}\}=\bigcup _{N}\bigcap _{n\geq N}A_{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=6" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li><a href="/wiki/Glossary_of_probability_and_statistics" title="Glossary of probability and statistics">Glossary of probability and statistics</a></li> <li><a href="/wiki/Combination" title="Combination">Combinations</a> and <a href="/wiki/Permutation" title="Permutation">permutations</a></li> <li><a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=7" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://analystprep.com/cfa-level-1-exam/quantitative-methods/calculating-probabilities-from-cumulative-distribution-function/">"Calculating Probabilities from Cumulative Distribution Function"</a>. 2021-08-09<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Calculating+Probabilities+from+Cumulative+Distribution+Function&rft.date=2021-08-09&rft_id=https%3A%2F%2Fanalystprep.com%2Fcfa-level-1-exam%2Fquantitative-methods%2Fcalculating-probabilities-from-cumulative-distribution-function%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANotation+in+probability+and+statistics" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1201/b15257-3">"Probability and stochastic processes"</a>, <i>Applied Stochastic Processes</i>, Chapman and Hall/CRC, pp. 9–36, 2013-07-22, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2Fb15257-3">10.1201/b15257-3</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-16812-3" title="Special:BookSources/978-0-429-16812-3"><bdi>978-0-429-16812-3</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2023-12-08</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Stochastic+Processes&rft.atitle=Probability+and+stochastic+processes&rft.pages=9-36&rft.date=2013-07-22&rft_id=info%3Adoi%2F10.1201%2Fb15257-3&rft.isbn=978-0-429-16812-3&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1201%2Fb15257-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANotation+in+probability+and+statistics" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://lesn.appstate.edu/olson/EDL7150/Components/Other%20useful%20links/Greek%20Alphabet%20and%20Statistics.htm">"Letters of the Greek Alphabet and Some of Their Statistical Uses"</a>. <i>les.appstate.edu/</i>. 1999-02-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=les.appstate.edu%2F&rft.atitle=Letters+of+the+Greek+Alphabet+and+Some+of+Their+Statistical+Uses&rft.date=1999-02-13&rft_id=https%3A%2F%2Flesn.appstate.edu%2Folson%2FEDL7150%2FComponents%2FOther%2520useful%2520links%2FGreek%2520Alphabet%2520and%2520Statistics.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANotation+in+probability+and+statistics" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf">"Order Statistics"</a> <span class="cs1-format">(PDF)</span>. <i>colorado.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=colorado.edu&rft.atitle=Order+Statistics&rft_id=https%3A%2F%2Fwww.colorado.edu%2Famath%2Fsites%2Fdefault%2Ffiles%2Fattached-files%2Forder_stats.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANotation+in+probability+and+statistics" class="Z3988"></span></span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalperinHartleyHoel1965" class="citation cs2">Halperin, Max; Hartley, H. O.; Hoel, P. G. (1965), "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation", <i>The American Statistician</i>, <b>19</b> (3): 12–14, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2681417">10.2307/2681417</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2681417">2681417</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Statistician&rft.atitle=Recommended+Standards+for+Statistical+Symbols+and+Notation.+COPSS+Committee+on+Symbols+and+Notation&rft.volume=19&rft.issue=3&rft.pages=12-14&rft.date=1965&rft_id=info%3Adoi%2F10.2307%2F2681417&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2681417%23id-name%3DJSTOR&rft.aulast=Halperin&rft.aufirst=Max&rft.au=Hartley%2C+H.+O.&rft.au=Hoel%2C+P.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANotation+in+probability+and+statistics" class="Z3988"></span></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Notation_in_probability_and_statistics&action=edit&section=8" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul><li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/stat.html">Earliest Uses of Symbols in Probability and Statistics</a>, maintained by Jeff Miller.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div>    </section></div> <noscript><img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Notation_in_probability_and_statistics&oldid=1256842542">https://en.wikipedia.org/w/index.php?title=Notation_in_probability_and_statistics&oldid=1256842542</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Notation_in_probability_and_statistics&action=history"> <div class="post-content last-modified-bar__content"> <span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Citation bot" data-user-gender="unknown" data-timestamp="1731358832"> <span>Last edited on 11 November 2024, at 21:00</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Nota%C3%A7%C3%A3o_em_probabilidade_e_estat%C3%ADstica" title="Notação em probabilidade e estatística – Portuguese" lang="pt" hreflang="pt" data-title="Notação em probabilidade e estatística" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 11 November 2024, at 21:00<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-terms-use"><a href="https://foundation.m.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">Terms of Use</a></li> <li id="footer-places-desktop-toggle"><a id="mw-mf-display-toggle" href="//en.wikipedia.org/w/index.php?title=Notation_in_probability_and_statistics&mobileaction=toggle_view_desktop" data-event-name="switch_to_desktop">Desktop</a></li> </ul> </div> </footer> </div> </div> <div class="mw-notification-area" data-mw="interface"></div>  <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-q49ns","wgBackendResponseTime":154,"wgPageParseReport":{"limitreport":{"cputime":"0.454","walltime":"0.648","ppvisitednodes":{"value":1803,"limit":1000000},"postexpandincludesize":{"value":70464,"limit":2097152},"templateargumentsize":{"value":689,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":56224,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 426.948 1 -total"," 30.26% 129.210 1 Template:Reflist"," 25.25% 107.814 3 Template:Cite_web"," 23.32% 99.545 1 Template:ProbabilityTopicsTOC"," 22.78% 97.269 2 Template:Sidebar"," 17.01% 72.603 1 Template:Short_description"," 13.47% 57.518 5 Template:Unreferenced_section"," 11.94% 50.996 5 Template:Unreferenced"," 11.48% 49.024 2 Template:Pagetype"," 10.86% 46.347 5 Template:Ambox"]},"scribunto":{"limitreport-timeusage":{"value":"0.255","limit":"10.000"},"limitreport-memusage":{"value":4706951,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\ntable#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-rnwnx","timestamp":"20241122150829","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Notation in probability and statistics","url":"https:\/\/en.wikipedia.org\/wiki\/Notation_in_probability_and_statistics","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1338010","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1338010","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-06-30T10:34:14Z","dateModified":"2024-11-11T21:00:32Z"}</script><script>(window.NORLQ=window.NORLQ||[]).push(function(){var ns,i,p,img;ns=document.getElementsByTagName('noscript');for(i=0;i<ns.length;i++){p=ns[i].nextSibling;if(p&&p.className&&p.className.indexOf('lazy-image-placeholder')>-1){img=document.createElement('img');img.setAttribute('src',p.getAttribute('data-src'));img.setAttribute('width',p.getAttribute('data-width'));img.setAttribute('height',p.getAttribute('data-height'));img.setAttribute('alt',p.getAttribute('data-alt'));p.parentNode.replaceChild(img,p);}}});</script> </body> </html>