Merge pull request #93 from kjpetrykowski/master

added [chi-2, chi-2-gam, mom-chi-2]

Author: JoramSoch

D/chi-2.md

@@ -0,0 +1,53 @@
+---
+layout: definition
+mathjax: true
+
+author: "Kenneth Petrykowski"
+affiliation: "KU Leuven"
+e_mail: "kenneth.petrykowski@gmail.com"
+date: 2020-10-13 01:20:00
+
+title: "Chi-square distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Chi-square distribution"
+definition: "Definition"
+
+sources:
+ - authors: "Wikipedia"
+   year: 2020
+   title: "Exponential distribution"
+   in: "Wikipedia, the free encyclopedia"
+   pages: "retrieved on 2020-10-12"
+   url: "https://en.wikipedia.org/wiki/Chi-square_distribution#Definitions"
+
+ - authors: "Robert V. Hogg, Joseph W. McKean, Allen T. Craig"
+   year: 2018
+   title: "The χ2-Distribution"
+   in: "Introduction to Mathematical Statistics"
+   pages: "Pearson, Boston, 2019, p. 178, eq. 3.3.7"
+   url: "https://www.pearson.com/store/p/introduction-to-mathematical-statistics/P100000843744"
+
+
+
+def_id: "D100"
+shortcut: "chi-2"
+username: "kjpetrykowski"
+---
+
+
+**Definition:** Let $X_{1}, ..., X_{k}$ be [independent random variables](/D/rvar), where each of them is following standard normal distribution  ($X_{i} \sim \mathcal N(0,1)$). Then, the sum of their squares,
+$Y\ =\sum _{i=1}^{k}X_{i}^{2},$ follows  chi-square distribution with $k$ degrees of freedom 
+$$\label{eq:chi-2} Y\ \sim \ \chi ^{2}(k)\ ; , $$
+where $k > 0$.
+$
+
+**Definition:** Let $Y$ be a random continous variable. Then, $Y$ is said to follow a chi-square distribution with $k$ number of degress of freedom
+
+$$\label{eq:chi-2} Y\ \sim \ \chi ^{2}(k)\ ; ,$$
+
+if and only if its probability density function is given by
+
+$$ \label{eq:chi-2-pdf} \chi ^{2}(x; k) = {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\; $$
+
+where $k > 0$ and the density is zero if $x \leq 0$.

P/chi-2-gam.md

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+---
+
+layout: proof
+mathjax: true
+
+author: "Kenneth Petrykowski"
+affiliation: "Ku Leuven"
+e_mail: "kenneth.petrykowski@gmail.com"
+date: 2020-10-12 22:15:00
+
+title: "Relationship between gamma and chi-squared distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Chi-square distribution"
+theorem: "Relationship between gamma and chi-squared distribution"
+
+sources:
+
+proof_id: "P174"
+shortcut: "chi-2-gam"
+username: "kjpetrykowski"
+---
+
+
+**Theorem:** The [chi-square distribution](/D/chi-2) with $k$ degrees of freedom $X\sim \chi _{k}^{2}$ is a special case of the [gamma distribution](/D/gam) that $X\sim \Gamma \left({\frac {k}{2}},{\frac {1}{2}}\right) $
+
+
+**Proof:** The [probability density function of the gamma distribution](/P/gam-pdf) for $x > 0$,  where $\alpha$ is the shape parameter and $\beta$ is the rate paramete, is as follows:
+
+$$ \label{eq:gamm-pdf}
+{ \mathrm{Gam}(x; \alpha, \beta) =\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}
+$$
+
+If we let $\alpha = k/2$ and $\beta = 1/2$ we obtain:
+
+$$
+{ \mathrm{Gam}(x; \frac{k}{2}, \frac{1}{2}) = \frac{x ^ {\frac{k}{2} - 1} e ^ {-x / 2}}{\Gamma(k / 2) 2 ^ {k / 2}} = {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\;} 
+$$\
+
+which is equivalent to the [probability density function of the chi-square distribution](/D/chi-2).

P/mom-chi-2.md

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+---
+
+layout: proof
+mathjax: true
+
+author: "Kenneth Petrykowski"
+affiliation: "Ku Leuven"
+e_mail: "kenneth.petrykowski@gmail.com"
+date: 2020-10-13 01:30:00
+
+title: "M-th moment of chi square distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Chi-square distribution"
+theorem: "M-th moment of chi square distribution"
+
+sources:
+ - authors: "Robert V. Hogg, Joseph W. McKean, Allen T. Craig"
+   year: 2018
+   title: "The χ2-Distribution"
+   in: "Introduction to Mathematical Statistics"
+   pages: "Pearson, Boston, 2019, p. 179, eq. 3.3.8"
+   url: "https://www.pearson.com/store/p/introduction-to-mathematical-statistics/P100000843744"
+
+
+
+proof_id: "P175"
+shortcut: "mom-chi-2"
+username: "kjpetrykowski"
+---
+
+
+**Theorem:** Let $X$ have a $\chi^{2}(k)$ distribution. If $m > -k/2$, then $E(X^{m})$ exists and is equal to:
+$$ \label{eq:chi-2-mom}
+{ {E}(X^{m}) =\frac {2^{m} \Gamma (\frac{k}{2} +m)}{\Gamma(\frac{k}{2})}\;,}$$  if  $m  >  -k/2$. \
+
+**Proof:** Observe that:
+
+$$ \label{eq:chi-2-mom-int}
+{ {E}(X^{m}) = \int_{0}^{\infty} {\frac {1}{\Gamma (\frac{k}{2}) 2^{k/2}}}\;x^{(k/2)+m-1}e^{-x/2}dx\;}. 
+$$
+
+Now set a new variable $u$ that $u=x/2$. As a result, we obtain:
+
+$$ \label{eq:chi-2-mom-int-u}
+{ {E}(X^{m}) = \int_{0}^{\infty} {\frac {1}{\Gamma (\frac{k}{2}) 2^{(k/2)-1}}}\;2^{(k/2)+m-1}u^{(k/2)+m-1}e^{-u}du\;}. 
+$$
+
+This leads to the desired result when $m$ $> -\frac{k}{2}$ \
+Observe that if $m$ is a nonnegative integer, then $m$ $> -\frac{k}{2}$ is always true. Therefore,
+all moments of a chi-square distribution exist and the m-th moment is given by this foregoing equation.