integrated new content

Definition "chi2" and proofs "chi2-gam"/"chi2-mom" were integrated into the archive.

Author: JoramSoch

D/chi-2.md D/chi2.mdD/chi-2.md renamed to D/chi2.md

@@ -16,38 +16,39 @@ definition: "Definition"
 sources:
  - authors: "Wikipedia"
    year: 2020
-   title: "Exponential distribution"
+   title: "Chi-square 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"
+shortcut: "chi2"
 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 $X_{1}, ..., X_{k}$ be [independent](/D/ind) [random variables](/D/rvar) where each of them is following a [standard normal distribution](/D/snorm):
+
+$$ \label{eq:snorm}
+X_{i} \sim \mathcal{N}(0,1) \; .
+$$.
 
-**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
+Then, the sum of their squares follows a chi-square distribution with $k$ degrees of freedom:
 
-$$\label{eq:chi-2} Y\ \sim \ \chi ^{2}(k)\ ; ,$$
+$$\label{eq:chi2}
+Y = \sum_{i=1}^{k} X_{i}^{2} \sim \chi^{2}(k) \quad \text{where} \quad k > 0 \; .
+$$
 
-if and only if its probability density function is given by
+A [random variables](/D/rvar) $Y$ is said said to follow a chi-square distribution with $k$ number of degress of freedom, if and only if its [probability density function](/D/pdf) is given by
 
-$$ \label{eq:chi-2-pdf} \chi ^{2}(x; k) = {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{k/2-1}e^{-x/2}\; $$
+$$ \label{eq:chi2-pdf}
+\chi^{2}(x; k) = \frac{1}{2^{k/2}\Gamma (k/2)} \, x^{k/2-1} \, e^{-x/2}
+$$
 
 where $k > 0$ and the density is zero if $x \leq 0$.

P/chi-2-gam.md

@@ -0,0 +1,43 @@
+---
+layout: proof
+mathjax: true
+
+author: "Kenneth Petrykowski"
+affiliation: "KU Leuven"
+e_mail: "kenneth.petrykowski@gmail.com"
+date: 2020-10-12 22:15:00
+
+title: "Chi-square distribution is a special case of gamma distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Chi-square distribution"
+theorem: "Special case of gamma distribution"
+
+sources:
+
+proof_id: "P174"
+shortcut: "chi2-gam"
+username: "kjpetrykowski"
+---
+
+
+**Theorem:** The [chi-square distribution](/D/chi2) with $k$ degrees of freedom is a special case of the [gamma distribution](/D/gam) with shape $\frac{k}{2}$ and rate $\frac{1}{2}$:
+
+$$ \label{eq:chi2-gam}
+X \sim \mathrm{Gam}\left( \frac{k}{2}, \frac{1}{2} \right) \Rightarrow X \sim \chi^{2}(k) \; .
+$$
+
+
+**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:
+
+$$ \label{eq:gam-pdf}
+\mathrm{Gam}(x; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \, x^{\alpha-1} \, e^{-\beta x}
+$$
+
+If we let $\alpha = k/2$ and $\beta = 1/2$, we obtain
+
+$$ \label{eq:gam-pdf-chi2}
+\mathrm{Gam}\left(x; \frac{k}{2}, \frac{1}{2}\right) = \frac{x^{k/2-1} \, e^{-x/2}}{\Gamma(k/2) 2^{k/2}} = \frac{1}{2^{k/2} \Gamma(k/2)} \, x^{k/2-1} \, e^{-x/2}
+$$\
+
+which is equivalent to the [probability density function of the chi-square distribution](/P/chi2-pdf).

P/chi2-gam.md

@@ -0,0 +1,55 @@
+---
+layout: proof
+mathjax: true
+
+author: "Kenneth Petrykowski"
+affiliation: "KU Leuven"
+e_mail: "kenneth.petrykowski@gmail.com"
+date: 2020-10-13 01:30:00
+
+title: "Moments of the chi-square distribution"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Chi-square distribution"
+theorem: "Moments"
+
+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: "chi2-mom"
+username: "kjpetrykowski"
+---
+
+
+**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [chi-square distribution](/D/chi2):
+
+$$ \label{eq:chi2}
+X \sim \chi^{2}(k) \; .
+$$
+
+If $m > -k/2$, then $E(X^{m})$ exists and is equal to:
+
+$$ \label{eq:chi2-mom}
+\mathrm{E}(X^{m}) = \frac{2^{m} \Gamma\left( \frac{k}{2}+m \right)}{\Gamma\left( \frac{k}{2} \right)} \; .
+$$
+
+
+**Proof:** Combining the [definition of the $m$-th raw moment](/D/momraw) with the [probability density function of the chi-square distribution](/P/chi2-pdf), we have:
+
+$$ \label{eq:chi2-mom-int}
+\mathrm{E}(X^{m}) = \int_{0}^{\infty} \frac{1}{\Gamma\left( \frac{k}{2} \right) 2^{k/2}} \, x^{(k/2)+m-1} \, e^{-x/2} \mathrm{d}x \; . 
+$$
+
+Now define a new variable $u = x/2$. As a result, we obtain:
+
+$$ \label{eq:chi-2-mom-int-u}
+\mathrm{E}(X^{m}) = \int_{0}^{\infty} \frac{1}{\Gamma\left( \frac{k}{2} \right) 2^{(k/2)-1}} \, 2^{(k/2)+m-1} \, u^{(k/2)+m-1} \, e^{-u} \mathrm{d}u \; .
+$$
+
+This leads to the desired result when $m > -k/2$. Observe that, if $m$ is a nonnegative integer, then $m > -k/2$ is always true. Therefore, all [moments](/D/mom) of a [chi-square distribution](/D/chi2) exist and the $m$-th raw moment is given by the foregoing equation.