Exchangeable random variables

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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Concept in statistics</div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, an <b>exchangeable sequence of random variables</b> (also sometimes <b>interchangeable</b>)<sup id="cite_ref-ChowTeicher_1-0" class="reference"><a href="#cite_note-ChowTeicher-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a sequence <i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, <i>X</i><sub>3</sub>, ... (which may be finitely or infinitely long) whose <a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences </p> <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_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\quad {\text{ and }}\quad X_{3},X_{6},X_{1},X_{5},X_{2},X_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" 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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <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>5</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\quad {\text{ and }}\quad X_{3},X_{6},X_{1},X_{5},X_{2},X_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251087dbc3f2c2014f3d49f0bcb25959c52c6a51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:55.635ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},X_{4},X_{5},X_{6}\quad {\text{ and }}\quad X_{3},X_{6},X_{1},X_{5},X_{2},X_{4}}"></span></dd></dl> <p>both have the same joint probability distribution. </p><p>It is closely related to the use of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed random variables</a> in statistical models. Exchangeable sequences of random variables arise in cases of <a href="/wiki/Simple_random_sampling" class="mw-redirect" title="Simple random sampling">simple random sampling</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, an <b>exchangeable sequence of random variables</b> is a finite or infinite sequence <i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, <i>X</i><sub>3</sub>, ... of <a href="/wiki/Random_variable" title="Random variable">random variables</a> such that for any finite <a href="/wiki/Permutation" title="Permutation">permutation</a> σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the <a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">joint probability distribution</a> of the permuted sequence </p> <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_{\sigma (1)},X_{\sigma (2)},X_{\sigma (3)},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\sigma (1)},X_{\sigma (2)},X_{\sigma (3)},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7577de3143dc6f9ce3b0d0701f4823c9f3c7ecc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.419ex; height:3.009ex;" alt="{\displaystyle X_{\sigma (1)},X_{\sigma (2)},X_{\sigma (3)},\dots }"></span></dd></dl> <p>is the same as the joint probability distribution of the original sequence.<sup id="cite_ref-ChowTeicher_1-1" class="reference"><a href="#cite_note-ChowTeicher-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> </p><p>(A sequence <i>E</i><sub>1</sub>, <i>E</i><sub>2</sub>, <i>E</i><sub>3</sub>, ... of events is said to be exchangeable precisely if the sequence of its <a href="/wiki/Indicator_function" title="Indicator function">indicator functions</a> is exchangeable.) The distribution function <i>F</i><sub><i>X</i><sub>1</sub>,...,<i>X</i><sub><i>n</i></sub></sub>(<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>) of a finite sequence of exchangeable random variables is symmetric in its arguments <span class="nowrap"><i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>.</span> <a href="/wiki/Olav_Kallenberg" title="Olav Kallenberg">Olav Kallenberg</a> provided an appropriate definition of exchangeability for continuous-time stochastic processes.<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><sup id="cite_ref-Kallenberg_4-0" class="reference"><a href="#cite_note-Kallenberg-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept was introduced by <a href="/wiki/William_Ernest_Johnson" title="William Ernest Johnson">William Ernest Johnson</a> in his 1924 book <i>Logic, Part III: The Logical Foundations of Science</i>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Exchangeability is equivalent to the concept of <a href="/wiki/Statistical_control" class="mw-redirect" title="Statistical control">statistical control</a> introduced by <a href="/wiki/Walter_Shewhart" class="mw-redirect" title="Walter Shewhart">Walter Shewhart</a> also in 1924.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Exchangeability_and_the_i.i.d._statistical_model">Exchangeability and the i.i.d. statistical model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=3" title="Edit section: Exchangeability and the i.i.d. statistical model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The property of exchangeability is closely related to the use of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent and identically distributed</a> (i.i.d.) random variables in statistical models.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> A sequence of random variables that are i.i.d, conditional on some underlying distributional form, is exchangeable. This follows directly from the structure of the joint probability distribution generated by the i.i.d. form. </p><p>Mixtures of exchangeable sequences (in particular, sequences of i.i.d. variables) are exchangeable. The converse can be established for infinite sequences, through an important <a href="/wiki/De_Finetti%27s_theorem" title="De Finetti's theorem">representation theorem</a> by <a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a> (later extended by other probability theorists such as <a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos</a> and <a href="/wiki/Leonard_Jimmie_Savage" title="Leonard Jimmie Savage">Savage</a>).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The extended versions of the theorem show that in any infinite sequence of exchangeable random variables, the random variables are conditionally <a href="/wiki/Independent_and_identically-distributed_random_variables" class="mw-redirect" title="Independent and identically-distributed random variables">independent and identically-distributed</a>, given the underlying distributional form. This theorem is stated briefly below. (De Finetti's original theorem only showed this to be true for random indicator variables, but this was later extended to encompass all sequences of random variables.) Another way of putting this is that <a href="/wiki/De_Finetti%27s_theorem" title="De Finetti's theorem">de Finetti's theorem</a> characterizes exchangeable sequences as mixtures of i.i.d. sequences—while an exchangeable sequence need not itself be unconditionally i.i.d., it can be expressed as a mixture of underlying i.i.d. sequences.<sup id="cite_ref-ChowTeicher_1-2" class="reference"><a href="#cite_note-ChowTeicher-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>This means that infinite sequences of exchangeable random variables can be regarded equivalently as sequences of conditionally i.i.d. random variables, based on some underlying distributional form. (Note that this equivalence does not quite hold for finite exchangeability. However, for finite vectors of random variables there is a close approximation to the i.i.d. model.) An infinite exchangeable sequence is <a href="/wiki/Strictly_stationary" class="mw-redirect" title="Strictly stationary">strictly stationary</a> and so a <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> in the form of <a href="/wiki/Birkhoff%E2%80%93Khinchin_theorem" class="mw-redirect" title="Birkhoff–Khinchin theorem">Birkhoff–Khinchin theorem</a> applies.<sup id="cite_ref-Kallenberg_4-1" class="reference"><a href="#cite_note-Kallenberg-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This means that the underlying distribution can be given an operational interpretation as the limiting empirical distribution of the sequence of values. The close relationship between exchangeable sequences of random variables and the i.i.d. form means that the latter can be justified on the basis of infinite exchangeability. This notion is central to <a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti's</a> development of <a href="/wiki/Predictive_inference" class="mw-redirect" title="Predictive inference">predictive inference</a> and to <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>. It can also be shown to be a useful foundational assumption in <a href="/wiki/Frequentist_statistics" class="mw-redirect" title="Frequentist statistics">frequentist statistics</a> and to link the two paradigms.<sup id="cite_ref-O'Neill_10-0" class="reference"><a href="#cite_note-O'Neill-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><b>The representation theorem:</b> This statement is based on the presentation in O'Neill (2009) in references below. Given an infinite sequence of random variables <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 \mathbf {X} =(X_{1},X_{2},X_{3},\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},X_{2},X_{3},\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be35f75388cd2f72e34fb218afa73a26b864b0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.688ex; height:2.843ex;" alt="{\displaystyle \mathbf {X} =(X_{1},X_{2},X_{3},\ldots )}"></span> we define the limiting <a href="/wiki/Empirical_distribution_function" title="Empirical distribution function">empirical distribution function</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 F_{\mathbf {X} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a05c045f0196d825382a50f4f4bd9d2a7dd0e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.155ex; height:2.509ex;" alt="{\displaystyle F_{\mathbf {X} }}"></span> by </p> <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 F_{\mathbf {X} }(x)=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}I(X_{i}\leq x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>I</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }(x)=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}I(X_{i}\leq x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c320d1d6fe2aff9b7ee30ec42eebaf5dfdf0f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.192ex; height:6.843ex;" alt="{\displaystyle F_{\mathbf {X} }(x)=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}I(X_{i}\leq x).}"></span></dd></dl> <p>(This is the <a href="/wiki/Ces%C3%A0ro_limit" class="mw-redirect" title="Cesàro limit">Cesàro limit</a> of the indicator functions. In cases where the Cesàro limit does not exist this function can actually be defined as the <a href="/wiki/Banach_limit" title="Banach limit">Banach limit</a> of the indicator functions, which is an extension of this limit. This latter limit always exists for sums of indicator functions, so that the empirical distribution is always well-defined.) This means that for any vector of random variables in the sequence we have joint distribution function given by </p> <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 \Pr(X_{1}\leq x_{1},X_{2}\leq x_{2},\ldots ,X_{n}\leq x_{n})=\int \prod _{i=1}^{n}F_{\mathbf {X} }(x_{i})\,dP(F_{\mathbf {X} }).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <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>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</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> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(X_{1}\leq x_{1},X_{2}\leq x_{2},\ldots ,X_{n}\leq x_{n})=\int \prod _{i=1}^{n}F_{\mathbf {X} }(x_{i})\,dP(F_{\mathbf {X} }).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/811f1515e82cc43295bbe74f016a164cf0d8cedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.316ex; height:6.843ex;" alt="{\displaystyle \Pr(X_{1}\leq x_{1},X_{2}\leq x_{2},\ldots ,X_{n}\leq x_{n})=\int \prod _{i=1}^{n}F_{\mathbf {X} }(x_{i})\,dP(F_{\mathbf {X} }).}"></span></dd></dl> <p>If the distribution function <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_{\mathbf {X} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathbf {X} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a05c045f0196d825382a50f4f4bd9d2a7dd0e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.155ex; height:2.509ex;" alt="{\displaystyle F_{\mathbf {X} }}"></span> is indexed by another parameter <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><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 }"></span> then (with densities appropriately defined) we have </p> <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 p_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\int \prod _{i=1}^{n}p_{X_{i}}(x_{i}\mid \theta )\,dP(\theta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\int \prod _{i=1}^{n}p_{X_{i}}(x_{i}\mid \theta )\,dP(\theta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056df64403939ed9984a6484414841d16ea72687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:47.053ex; height:6.843ex;" alt="{\displaystyle p_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\int \prod _{i=1}^{n}p_{X_{i}}(x_{i}\mid \theta )\,dP(\theta ).}"></span></dd></dl> <p>These equations show the joint distribution or density characterised as a mixture distribution based on the underlying limiting empirical distribution (or a parameter indexing this distribution). </p><p>Note that not all finite exchangeable sequences are mixtures of i.i.d. To see this, consider sampling without replacement from a finite set until no elements are left. The resulting sequence is exchangeable, but not a mixture of i.i.d. Indeed, conditioned on all other elements in the sequence, the remaining element is known. </p> <div class="mw-heading mw-heading2"><h2 id="Covariance_and_correlation">Covariance and correlation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=4" title="Edit section: Covariance and correlation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Exchangeable sequences have some basic covariance and correlation properties which mean that they are generally positively correlated. For infinite sequences of exchangeable random variables, the covariance between the random variables is equal to the variance of the mean of the underlying distribution function.<sup id="cite_ref-O'Neill_10-1" class="reference"><a href="#cite_note-O'Neill-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> For finite exchangeable sequences the covariance is also a fixed value which does not depend on the particular random variables in the sequence. There is a weaker lower bound than for infinite exchangeability and it is possible for negative correlation to exist. </p><p><b>Covariance for exchangeable sequences (infinite):</b> If the sequence <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},X_{2},X_{3},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" 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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1dd59b3860ca16ef451f9000c769aaa3ce593b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.761ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},\ldots }"></span> is exchangeable, then </p> <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 \operatorname {cov} (X_{i},X_{j})=\operatorname {var} (\operatorname {E} (X_{i}\mid F_{\mathbf {X} }))=\operatorname {var} (\operatorname {E} (X_{i}\mid \theta ))\geq 0\quad {\text{for }}i\neq j.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∣</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X_{i},X_{j})=\operatorname {var} (\operatorname {E} (X_{i}\mid F_{\mathbf {X} }))=\operatorname {var} (\operatorname {E} (X_{i}\mid \theta ))\geq 0\quad {\text{for }}i\neq j.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e16eb6b728b12b4fa27f75343e1a2b47dd7af191" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:64.05ex; height:3.009ex;" alt="{\displaystyle \operatorname {cov} (X_{i},X_{j})=\operatorname {var} (\operatorname {E} (X_{i}\mid F_{\mathbf {X} }))=\operatorname {var} (\operatorname {E} (X_{i}\mid \theta ))\geq 0\quad {\text{for }}i\neq j.}"></span></dd></dl> <p><b>Covariance for exchangeable sequences (finite):</b> 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 X_{1},X_{2},\ldots ,X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" 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">{\displaystyle X_{1},X_{2},\ldots ,X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67872301909a9d739e265252ad0c7339cead069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}"></span> is exchangeable 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="{\displaystyle \sigma ^{2}=\operatorname {var} (X_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=\operatorname {var} (X_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9c06df0d7e81b3b897a5b24f444887c177970d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}=\operatorname {var} (X_{i})}"></span>, then </p> <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 \operatorname {cov} (X_{i},X_{j})\geq -{\frac {\sigma ^{2}}{n-1}}\quad {\text{for }}i\neq j.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (X_{i},X_{j})\geq -{\frac {\sigma ^{2}}{n-1}}\quad {\text{for }}i\neq j.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfafc5e988c768cf94c8fbb7cc987423226ed30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.159ex; height:5.843ex;" alt="{\displaystyle \operatorname {cov} (X_{i},X_{j})\geq -{\frac {\sigma ^{2}}{n-1}}\quad {\text{for }}i\neq j.}"></span></dd></dl> <p>The finite sequence result may be proved as follows. Using the fact that the values are exchangeable, we have </p> <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 {\begin{aligned}0&\leq \operatorname {var} (X_{1}+\cdots +X_{n})\\&=\operatorname {var} (X_{1})+\cdots +\operatorname {var} (X_{n})+\underbrace {\operatorname {cov} (X_{1},X_{2})+\cdots \quad {}} _{\text{all ordered pairs}}\\&=n\sigma ^{2}+n(n-1)\operatorname {cov} (X_{1},X_{2}).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <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 stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>all ordered pairs</mtext> </mrow> </munder> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>cov</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <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 stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}0&\leq \operatorname {var} (X_{1}+\cdots +X_{n})\\&=\operatorname {var} (X_{1})+\cdots +\operatorname {var} (X_{n})+\underbrace {\operatorname {cov} (X_{1},X_{2})+\cdots \quad {}} _{\text{all ordered pairs}}\\&=n\sigma ^{2}+n(n-1)\operatorname {cov} (X_{1},X_{2}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af27df1b51c3135258b35e740ae7a0eb45d5ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:53.096ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}0&\leq \operatorname {var} (X_{1}+\cdots +X_{n})\\&=\operatorname {var} (X_{1})+\cdots +\operatorname {var} (X_{n})+\underbrace {\operatorname {cov} (X_{1},X_{2})+\cdots \quad {}} _{\text{all ordered pairs}}\\&=n\sigma ^{2}+n(n-1)\operatorname {cov} (X_{1},X_{2}).\end{aligned}}}"></span></dd></dl> <p>We can then solve the inequality for the covariance yielding the stated lower bound. The non-negativity of the covariance for the infinite sequence can then be obtained as a limiting result from this finite sequence result. </p><p>Equality of the lower bound for finite sequences is achieved in a simple urn model: An urn contains 1 red marble and <i>n</i> − 1 green marbles, and these are sampled without replacement until the urn is empty. Let <i>X</i><sub><i>i</i></sub> = 1 if the red marble is drawn on the <i>i</i>-th trial and 0 otherwise. A finite sequence that achieves the lower covariance bound cannot be extended to a longer exchangeable sequence.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=5" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Any <a href="/wiki/Convex_combination" title="Convex combination">convex combination</a> or <a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture distribution</a> of <a href="/wiki/Iid" class="mw-redirect" title="Iid">iid</a> sequences of random variables is exchangeable. A converse proposition is <a href="/wiki/De_Finetti%27s_theorem" title="De Finetti's theorem">de Finetti's theorem</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>Suppose an <a href="/wiki/Urn_model" class="mw-redirect" title="Urn model">urn</a> contains <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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="{\displaystyle n}"></span> red 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> blue marbles. Suppose marbles are drawn without replacement until the urn is empty. Let <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> be the indicator random variable of the event that 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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th marble drawn is red. Then <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 \left\{X_{i}\right\}_{i=1,\dots ,n+m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{i}\right\}_{i=1,\dots ,n+m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb80c9dce0fad112f8e73aa6eade351d0549c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.497ex; height:3.176ex;" alt="{\displaystyle \left\{X_{i}\right\}_{i=1,\dots ,n+m}}"></span> is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.</li> <li>Suppose an urn contains <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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="{\displaystyle n}"></span> red 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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> blue marbles. Further suppose a marble is drawn from the urn and then replaced, with an extra marble of the same colour. Let <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}"></span> be the indicator random variable of the event that 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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>-th marble drawn is red. Then <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 \left\{X_{i}\right\}_{i\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{i}\right\}_{i\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4bc0c389acd750cb78b913e6b413dcbc4694a7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.131ex; height:3.009ex;" alt="{\displaystyle \left\{X_{i}\right\}_{i\in \mathbb {N} }}"></span> is an exchangeable sequence. This model is called <a href="/wiki/Polya%27s_urn" class="mw-redirect" title="Polya's urn">Polya's urn</a>.</li> <li>Let <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,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41f29b9537685f499713112d6802e811cbf51bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.597ex; height:2.843ex;" alt="{\displaystyle (X,Y)}"></span> have a <a href="/wiki/Bivariate_normal_distribution" class="mw-redirect" title="Bivariate normal distribution">bivariate normal distribution</a> with parameters <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 \mu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}"></span>, <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 \sigma _{x}=\sigma _{y}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}=\sigma _{y}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fceaba4da86a3ee484e8de3fc865dbaa2de6cfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.236ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}=\sigma _{y}=1}"></span> and an arbitrary <a href="/wiki/Pearson_product-moment_correlation_coefficient" class="mw-redirect" title="Pearson product-moment correlation coefficient">correlation coefficient</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 \rho \in (-1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \in (-1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05d5ab2639e435b4027d8296a803b66aa746263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.019ex; height:2.843ex;" alt="{\displaystyle \rho \in (-1,1)}"></span>. The random variables <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><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}"></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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" 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="{\displaystyle Y}"></span> are then exchangeable, but independent only 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 \rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba6310b27df5f9c9b0b1732e08cce27b99d68cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}"></span>. The <a href="/wiki/Density_function" class="mw-redirect" title="Density function">density function</a> is <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)=p(y,x)\propto \exp \left[-{\frac {1}{2(1-\rho ^{2})}}(x^{2}+y^{2}-2\rho xy)\right].}"> <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> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>ρ</mi> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x,y)=p(y,x)\propto \exp \left[-{\frac {1}{2(1-\rho ^{2})}}(x^{2}+y^{2}-2\rho xy)\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b775ed0d79b8bd7ece01f18b919cee7985f067b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.089ex; width:54.75ex; height:6.343ex;" alt="{\displaystyle p(x,y)=p(y,x)\propto \exp \left[-{\frac {1}{2(1-\rho ^{2})}}(x^{2}+y^{2}-2\rho xy)\right].}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=6" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Von_Neumann_extractor" class="mw-redirect" title="Von Neumann extractor">von Neumann extractor</a> is a <a href="/wiki/Randomness_extractor" title="Randomness extractor">randomness extractor</a> that depends on exchangeability: it gives a method to take an exchangeable sequence of 0s and 1s (<a href="/wiki/Bernoulli_trials" class="mw-redirect" title="Bernoulli trials">Bernoulli trials</a>), with some probability <i>p</i> of 0 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 q=1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=1-p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e387fe24ba3da5f9a0dc424923cdfc2c08990c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.34ex; height:2.509ex;" alt="{\displaystyle q=1-p}"></span> of 1, and produce a (shorter) exchangeable sequence of 0s and 1s with probability 1/2. </p><p>Partition the sequence into non-overlapping pairs: if the two elements of the pair are equal (00 or 11), discard it; if the two elements of the pair are unequal (01 or 10), keep the first. This yields a sequence of Bernoulli trials 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="{\displaystyle p=1/2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1/2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a23a9f66dbc9074ef2f2683ecd6b5c5721b759fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.492ex; height:2.843ex;" alt="{\displaystyle p=1/2,}"></span> as, by exchangeability, the odds of a given pair being 01 or 10 are equal. </p><p>Exchangeable random variables arise in the study of <a href="/wiki/U_statistic" class="mw-redirect" title="U statistic">U statistics</a>, particularly in the Hoeffding decomposition.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Exchangeability is a key assumption of the distribution-free inference method of <a href="/wiki/Conformal_prediction" title="Conformal prediction">conformal prediction</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/De_Finetti_theorem" class="mw-redirect" title="De Finetti theorem">De Finetti theorem</a></li> <li><a href="/wiki/Hewitt-Savage_zero-one_law" class="mw-redirect" title="Hewitt-Savage zero-one law">Hewitt-Savage zero-one law</a></li> <li><a href="/wiki/Resampling_(statistics)" title="Resampling (statistics)">Resampling</a></li> <li><a href="/wiki/Resampling_(statistics)#Permutation_tests" title="Resampling (statistics)">Resampling (statistics) § Permutation tests</a>, statistical tests based on exchanging between groups</li></ul> <div class="mw-heading mw-heading2"><h2 id="Refererences">Refererences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=8" title="Edit section: Refererences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 mw-references-columns"><ol class="references"> <li id="cite_note-ChowTeicher-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-ChowTeicher_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ChowTeicher_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-ChowTeicher_1-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">In short, the order of the sequence of random variables does not affect its joint probability distribution. <ul><li>Chow, Yuan Shih and Teicher, Henry, <i>Probability theory. Independence, interchangeability, martingales,</i> Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp. <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><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98228-0" title="Special:BookSources/0-387-98228-0">0-387-98228-0</a></li></ul> </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">Aldous, David J., <i>Exchangeability and related topics</i>, in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp. 1–198, Springer, Berlin, 1985. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-15203-3" title="Special:BookSources/978-3-540-15203-3">978-3-540-15203-3</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0099421">10.1007/BFb0099421</a></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 id="CITEREFDiaconis2009" class="citation journal cs1"><a href="/wiki/Persi_Diaconis" title="Persi Diaconis">Diaconis, Persi</a> (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-09-01262-2">"Book review: <i>Probabilistic symmetries and invariance principles</i> (Olav Kallenberg, Springer, New York, 2005)"</a>. <i>Bulletin of the American Mathematical Society</i>. 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"Predicting the unpredictable". <i>Synthese</i>. <b>90</b> (2): 205. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00485351">10.1007/bf00485351</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9416747">9416747</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Synthese&rft.atitle=Predicting+the+unpredictable&rft.volume=90&rft.issue=2&rft.pages=205&rft.date=1992&rft_id=info%3Adoi%2F10.1007%2Fbf00485351&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9416747%23id-name%3DS2CID&rft.aulast=Zabell&rft.aufirst=S.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Barlow, R. 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Oxford University Press. pp. 111–125. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-852220-7" title="Special:BookSources/0-19-852220-7"><bdi>0-19-852220-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Recent+Progress+on+de+Finetti%27s+Notions+of+Exchangeability&rft.btitle=Bayesian+Statistics&rft.pages=111-125&rft.pub=Oxford+University+Press&rft.date=1988&rft.isbn=0-19-852220-7&rft.aulast=Diaconis&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></span> </li> <li id="cite_note-O'Neill-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-O'Neill_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-O'Neill_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Neill2009" class="citation journal cs1">O'Neill, B. (2009). "Exchangeability, Correlation and Bayes' Effect". <i>International Statistical Review</i>. <b>77</b> (2): 241–250. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1751-5823.2008.00059.x">10.1111/j.1751-5823.2008.00059.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Statistical+Review&rft.atitle=Exchangeability%2C+Correlation+and+Bayes%27+Effect&rft.volume=77&rft.issue=2&rft.pages=241-250&rft.date=2009&rft_id=info%3Adoi%2F10.1111%2Fj.1751-5823.2008.00059.x&rft.aulast=O%27Neill&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylorDafferPatterson1985" class="citation book cs1">Taylor, Robert Lee; Daffer, Peter Z.; Patterson, Ronald F. (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6RaoAAAAIAAJ"><i>Limit theorems for sums of exchangeable random variables</i></a>. Rowman and Allanheld. pp. 1–152. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780847674350" title="Special:BookSources/9780847674350"><bdi>9780847674350</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Limit+theorems+for+sums+of+exchangeable+random+variables&rft.pages=1-152&rft.pub=Rowman+and+Allanheld&rft.date=1985&rft.isbn=9780847674350&rft.aulast=Taylor&rft.aufirst=Robert+Lee&rft.au=Daffer%2C+Peter+Z.&rft.au=Patterson%2C+Ronald+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6RaoAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Spizzichino, Fabio <i>Subjective probability models for lifetimes</i>. Monographs on Statistics and Applied Probability, 91. <i>Chapman & Hall/CRC</i>, Boca Raton, FL, 2001. xx+248 pp. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58488-060-0" title="Special:BookSources/1-58488-060-0">1-58488-060-0</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorovskikh1996" class="citation book cs1">Borovskikh, Yu. V. (1996). "Chapter 10 Dependent variables". <i><span></span></i>U<i>-statistics in Banach spaces</i>. Utrecht: VSP. pp. 365–376. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-6764-200-2" title="Special:BookSources/90-6764-200-2"><bdi>90-6764-200-2</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1419498">1419498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+10+Dependent+variables&rft.btitle=U-statistics+in+Banach+spaces&rft.place=Utrecht&rft.pages=365-376&rft.pub=VSP&rft.date=1996&rft.isbn=90-6764-200-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1419498%23id-name%3DMR&rft.aulast=Borovskikh&rft.aufirst=Yu.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShaferVovk2008" class="citation journal cs1">Shafer, Glenn; Vovk, Vladimir (2008). <a rel="nofollow" class="external text" href="https://www.jmlr.org/papers/v9/shafer08a.html">"A Tutorial on Conformal Prediction"</a>. <i>Journal of Machine Learning Research</i>. <b>9</b>: 371–421.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Machine+Learning+Research&rft.atitle=A+Tutorial+on+Conformal+Prediction&rft.volume=9&rft.pages=371-421&rft.date=2008&rft.aulast=Shafer&rft.aufirst=Glenn&rft.au=Vovk%2C+Vladimir&rft_id=https%3A%2F%2Fwww.jmlr.org%2Fpapers%2Fv9%2Fshafer08a.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Exchangeable_random_variables&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Aldous, David J., <i>Exchangeability and related topics</i>, in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp. 1–198, Springer, Berlin, 1985. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-15203-3" title="Special:BookSources/978-3-540-15203-3">978-3-540-15203-3</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0099421">10.1007/BFb0099421</a></li> <li>Chow, Yuan Shih and Teicher, Henry, <i>Probability theory. Independence, interchangeability, martingales,</i> Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98228-0" title="Special:BookSources/0-387-98228-0">0-387-98228-0</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDawid2013" class="citation book cs1">Dawid, A. Philip (2013). "Exchangeability and its ramifications". In Damien, Paul; et al. (eds.). <i>Bayesian Theory and Applications</i>. Oxford University Press. pp. 19–30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-969560-7" title="Special:BookSources/978-0-19-969560-7"><bdi>978-0-19-969560-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Exchangeability+and+its+ramifications&rft.btitle=Bayesian+Theory+and+Applications&rft.pages=19-30&rft.pub=Oxford+University+Press&rft.date=2013&rft.isbn=978-0-19-969560-7&rft.aulast=Dawid&rft.aufirst=A.+Philip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></li> <li><a href="/wiki/Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, O.</a>, <i>Probabilistic symmetries and invariance principles</i>. Springer-Verlag, New York (2005). 510 pp. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-25115-4" title="Special:BookSources/0-387-25115-4">0-387-25115-4</a>.</li> <li>Kingman, J. F. C., <i>Uses of exchangeability</i>, Ann. Probability 6 (1978) 83–197 <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=494344">494344</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/2243211">2243211</a></li> <li>O'Neill, B. (2009) Exchangeability, Correlation and Bayes' Effect. <i>International Statistical Review</i> <b>77(2)</b>, pp. 241–250. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-15203-3" title="Special:BookSources/978-3-540-15203-3">978-3-540-15203-3</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1751-5823.2008.00059.x">10.1111/j.1751-5823.2008.00059.x</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTaylorDafferPatterson1985" class="citation book cs1">Taylor, Robert Lee; Daffer, Peter Z.; Patterson, Ronald F. (1985). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6RaoAAAAIAAJ"><i>Limit theorems for sums of exchangeable random variables</i></a>. Rowman and Allanheld. pp. 1–152. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780847674350" title="Special:BookSources/9780847674350"><bdi>9780847674350</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Limit+theorems+for+sums+of+exchangeable+random+variables&rft.pages=1-152&rft.pub=Rowman+and+Allanheld&rft.date=1985&rft.isbn=9780847674350&rft.aulast=Taylor&rft.aufirst=Robert+Lee&rft.au=Daffer%2C+Peter+Z.&rft.au=Patterson%2C+Ronald+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6RaoAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AExchangeable+random+variables" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=Exchangeable_random_variables&oldid=1255480599">https://en.wikipedia.org/w/index.php?title=Exchangeable_random_variables&oldid=1255480599</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Statistical_randomness" title="Category:Statistical randomness">Statistical randomness</a></li><li><a href="/wiki/Category:Types_of_probability_distributions" title="Category:Types of probability distributions">Types of probability distributions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 5 November 2024, at 04:29<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Exchangeable random variables - Wikipedia