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probability</a>)</p> <h2 id="measure_theory">Measure theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+space">measurable space</a>, <a class="existingWikiWord" href="/nlab/show/measurable+locale">measurable locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measure">measure</a>, <a class="existingWikiWord" href="/nlab/show/measure+space">measure space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a></p> </li> </ul> <h2 id="probability_theory">Probability theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+space">probability space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state">state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/states+in+AQFT+and+operator+algebra">in AQFT and operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/entropy">entropy</a>, <a class="existingWikiWord" href="/nlab/show/relative+entropy">relative entropy</a></p> </li> </ul> <h2 id="information_geometry">Information geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/information+geometry">information geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/information+metric">information metric</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wasserstein+metric">Wasserstein metric</a></p> </li> </ul> <h2 id="thermodynamics">Thermodynamics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+law+of+thermodynamics">second law of thermodynamics</a>, <a class="existingWikiWord" href="/nlab/show/generalized+second+law+of+theormodynamics">generalized second law</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ergodic+theory">ergodic theory</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riesz+representation+theorem">Riesz representation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Finetti%27s+theorem">de Finetti's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/law+of+large+numbers">law of large numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+extension+theorem">Kolmogorov extension theorem</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/machine+learning">machine learning</a>, <a class="existingWikiWord" href="/nlab/show/neural+networks">neural networks</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A <a class="existingWikiWord" href="/nlab/show/probability+distribution">probability distribution</a> on a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is called <strong>Gaussian</strong> or a <strong>normal distribution</strong> if it is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mfrac><msqrt><mrow><mi>det</mi><mi>A</mi></mrow></msqrt><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mo>⟨</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>A</mi><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>⟩</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mi>⋯</mi><mi>d</mi><msup><mi>x</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu_A \;\colon\; \vec x \mapsto \frac{\sqrt{det A}}{(2\pi)^{n/2}} \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right) \,d x^1 \cdots d x^n \,. </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/matrix">matrix</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo>−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -, A-\rangle</annotation></semantics></math> is a positive definite <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\det A</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,-\rangle</annotation></semantics></math> is the canonical <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><mi>det</mi><mi>A</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\det A}</annotation></semantics></math> is the coordinate of the <a class="existingWikiWord" href="/nlab/show/volume+element">volume element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>vol</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">vol_A</annotation></semantics></math> associated with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, we may equivalently write this as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>↦</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mrow><mo>⟨</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>A</mi><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>⟩</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><msub><mi>vol</mi> <mi>A</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu_A \;\colon\; \vec x \mapsto \frac{1}{(2\pi)^{n/2}} \exp\left(-\tfrac{1}{2} \left\langle \vec x , A \vec x\right\rangle\right) \,vol_A \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/mean">mean</a> of this distribution is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mn>0</mn><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{0}</annotation></semantics></math>; for a distribution with mean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{c}</annotation></semantics></math>, replace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mrow><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>A</mi><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle{\vec{x},A \vec{x}}\rangle</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mrow><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mover><mi>c</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>A</mi><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>−</mo><mi>A</mi><mover><mi>c</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle{\vec{x} - \vec{c},A \vec{x} - A \vec{c}}\rangle</annotation></semantics></math>.</p> <p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/inverse+matrix">inverse</a> of the <span class="newWikiWord">covariance matrix<a href="/nlab/new/covariance+matrix">?</a></span>. In particular, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>, we may write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>σ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">x^2/\sigma^2</annotation></semantics></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">/</mo><msup><mi>σ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(x-c)^2/\sigma^2</annotation></semantics></math> for mean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>) in place of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mrow><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>A</mi><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle{\vec{x},A \vec{x}}\rangle</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/standard+deviation">standard deviation</a>; similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><mi>det</mi><mi>A</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\det A}</annotation></semantics></math> becomes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mi>σ</mi></mrow><annotation encoding="application/x-tex">1/\sigma</annotation></semantics></math>. This gives the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>σ</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>↦</mo><mfrac><mn>1</mn><mrow><mi>σ</mi><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi> <mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mi>d</mi><mi>x</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mu_\sigma \;\colon\; x \mapsto \frac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x-c)^2}{2\sigma^2}\right) \,d x \,, </annotation></semantics></math></div> <p>which may be more familiar to some readers.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/statistical+significance">statistical significance</a></p> </li> </ul> <h2 id="references">References</h2> <p>See also</p> <ul> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Normal_distribution">Normal distribution</a></em></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Standard_deviation">Standard deviation</a></em></p> </li> <li> <p>Good Calculators, <em><a href="https://goodcalculators.com/standard-deviation-calculator/">Standard deviation calculator</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 23, 2020 at 08:56:35. 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