Merge pull request #171 from JoramSoch/master

added 1 proof

Author: JoramSoch

I/ToC.md

@@ -352,24 +352,25 @@ title: "Table of Contents"
    &emsp;&ensp; 3.2.5. **[Relationship to standard normal distribution](/P/norm-snorm3)** (3) <br>
    &emsp;&ensp; 3.2.6. **[Relationship to chi-squared distribution](/P/norm-chi2)** <br>
    &emsp;&ensp; 3.2.7. **[Relationship to t-distribution](/P/norm-t)** <br>
-   &emsp;&ensp; 3.2.8. **[Gaussian integral](/P/norm-gi)** <br>
-   &emsp;&ensp; 3.2.9. **[Probability density function](/P/norm-pdf)** <br>
-   &emsp;&ensp; 3.2.10. **[Moment-generating function](/P/norm-mgf)** <br>
-   &emsp;&ensp; 3.2.11. **[Cumulative distribution function](/P/norm-cdf)** <br>
-   &emsp;&ensp; 3.2.12. **[Cumulative distribution function without error function](/P/norm-cdfwerf)** <br>
-   &emsp;&ensp; 3.2.13. **[Probability of being within standard deviations from mean](/P/norm-probstd)** <br>
-   &emsp;&ensp; 3.2.14. **[Quantile function](/P/norm-qf)** <br>
-   &emsp;&ensp; 3.2.15. **[Mean](/P/norm-mean)** <br>
-   &emsp;&ensp; 3.2.16. **[Median](/P/norm-med)** <br>
-   &emsp;&ensp; 3.2.17. **[Mode](/P/norm-mode)** <br>
-   &emsp;&ensp; 3.2.18. **[Variance](/P/norm-var)** <br>
-   &emsp;&ensp; 3.2.19. **[Full width at half maximum](/P/norm-fwhm)** <br>
-   &emsp;&ensp; 3.2.20. **[Extreme points](/P/norm-extr)** <br>
-   &emsp;&ensp; 3.2.21. **[Inflection points](/P/norm-infl)** <br>
-   &emsp;&ensp; 3.2.22. **[Differential entropy](/P/norm-dent)** <br>
-   &emsp;&ensp; 3.2.23. **[Kullback-Leibler divergence](/P/norm-kl)** <br>
-   &emsp;&ensp; 3.2.24. **[Maximum entropy distribution](/P/norm-maxent)** <br>
-   &emsp;&ensp; 3.2.25. **[Linear combination](/P/norm-lincomb)** <br>
+   &emsp;&ensp; 3.2.8. **[Special case of multivariate normal distribution](/P/norm-mvn)** <br>
+   &emsp;&ensp; 3.2.9. **[Gaussian integral](/P/norm-gi)** <br>
+   &emsp;&ensp; 3.2.10. **[Probability density function](/P/norm-pdf)** <br>
+   &emsp;&ensp; 3.2.11. **[Moment-generating function](/P/norm-mgf)** <br>
+   &emsp;&ensp; 3.2.12. **[Cumulative distribution function](/P/norm-cdf)** <br>
+   &emsp;&ensp; 3.2.13. **[Cumulative distribution function without error function](/P/norm-cdfwerf)** <br>
+   &emsp;&ensp; 3.2.14. **[Probability of being within standard deviations from mean](/P/norm-probstd)** <br>
+   &emsp;&ensp; 3.2.15. **[Quantile function](/P/norm-qf)** <br>
+   &emsp;&ensp; 3.2.16. **[Mean](/P/norm-mean)** <br>
+   &emsp;&ensp; 3.2.17. **[Median](/P/norm-med)** <br>
+   &emsp;&ensp; 3.2.18. **[Mode](/P/norm-mode)** <br>
+   &emsp;&ensp; 3.2.19. **[Variance](/P/norm-var)** <br>
+   &emsp;&ensp; 3.2.20. **[Full width at half maximum](/P/norm-fwhm)** <br>
+   &emsp;&ensp; 3.2.21. **[Extreme points](/P/norm-extr)** <br>
+   &emsp;&ensp; 3.2.22. **[Inflection points](/P/norm-infl)** <br>
+   &emsp;&ensp; 3.2.23. **[Differential entropy](/P/norm-dent)** <br>
+   &emsp;&ensp; 3.2.24. **[Kullback-Leibler divergence](/P/norm-kl)** <br>
+   &emsp;&ensp; 3.2.25. **[Maximum entropy distribution](/P/norm-maxent)** <br>
+   &emsp;&ensp; 3.2.26. **[Linear combination](/P/norm-lincomb)** <br>
 
    3.3. t-distribution <br>
    &emsp;&ensp; 3.3.1. *[Definition](/D/t)* <br>

P/norm-mvn.md

@@ -0,0 +1,49 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-08-19 19:41:00
+
+title: "Normal distribution is a special case of multivariate normal distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Normal distribution"
+theorem: "Special case of matrix-normal distribution"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2022
+    title: "Multivariate normal distribution"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2022-08-19"
+    url: "https://en.wikipedia.org/wiki/Multivariate_normal_distribution"
+
+proof_id: "P331"
+shortcut: "norm-snorm"
+username: "JoramSoch"
+---
+
+
+**Theorem:** The [normal distribution](/D/norm) is a special case of the [multivariate normal distribution](/D/mvn) with number of variables $n = 1$, i.e. [random vector](/D/rvec) $x \in \mathbb{R}$, mean $\mu \in \mathbb{R}$ and covariance matrix $\Sigma = \sigma^2$.
+
+
+**Proof:** The [probability density function of the multivariate normal distribution](/P/mvn-pdf) is
+
+$$ \label{eq:mvn-pdf}
+\mathcal{N}(x; \mu, \Sigma) = \frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right] \; .
+$$
+
+Setting $n = 1$, such that $x, \mu \in \mathbb{R}$, and $\Sigma = \sigma^2$, we obtain
+
+$$ \label{eq:norm-pdf}
+\begin{split}
+\mathcal{N}(x; \mu, \sigma^2) &= \frac{1}{\sqrt{(2 \pi)^1 |\sigma^2|}} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} (\sigma^2)^{-1} (x-\mu) \right] \\
+&= \frac{1}{\sqrt{(2\pi) \sigma^2}} \cdot \exp\left[-\frac{1}{2 \sigma^2} (x-\mu)^2 \right] \\
+&= \frac{1}{\sqrt{2\pi} \sigma} \cdot \exp \left[ -\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2 \right]
+\end{split}
+$$
+
+which is equivalent to the [probability density function of the normal distribution](/P/norm-pdf).