Merge pull request #35 from StatProofBook/master

update to master

Author: JoramSoch

D/beta.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a beta distribution with shape parameters $\alpha$ and $\beta$
+**Definition:** Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a beta distribution with shape parameters $\alpha$ and $\beta$
 
 $$ \label{eq:beta}
 X \sim \mathrm{Bet}(\alpha, \beta) \; ,

D/chi2.md

@@ -0,0 +1,54 @@
+---
+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: "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: "chi2"
+username: "kjpetrykowski"
+---
+
+
+**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):
+
+$$ \label{eq:snorm}
+X_{i} \sim \mathcal{N}(0,1) \; .
+$$
+
+Then, the sum of their squares follows a chi-square distribution with $k$ degrees of freedom:
+
+$$\label{eq:chi2}
+Y = \sum_{i=1}^{k} X_{i}^{2} \sim \chi^{2}(k) \quad \text{where} \quad k > 0 \; .
+$$
+
+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
+
+$$ \label{eq:chi2-pdf}
+\chi^{2}(x; k) = \frac{1}{2^{k/2}\Gamma (k/2)} \, x^{k/2-1} \, e^{-x/2}
+$$
+
+where $k > 0$ and the density is zero if $x \leq 0$.

D/dir.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random vector](/D/rvec). Then, $X$ is said to follow a Dirichlet distribution with concentration parameters $\alpha = \left[ \alpha_1, \ldots, \alpha_k \right]$
+**Definition:** Let $X$ be a [random vector](/D/rvec). Then, $X$ is said to follow a Dirichlet distribution with concentration parameters $\alpha = \left[ \alpha_1, \ldots, \alpha_k \right]$
 
 $$ \label{eq:Dir}
 X \sim \mathrm{Dir}(\alpha) \; ,

D/dist-cond.md

@@ -27,4 +27,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the conditional distribution of $X$ given that $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ given that $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$. The conditional distribution of $X$ can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ and the [marginal distribution](/D/dist-marg) of $Y$ using the [law of conditional probability](/D/prob-cond).
+**Definition:** Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the conditional distribution of $X$ given that $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ given that $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$. The conditional distribution of $X$ can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ and the [marginal distribution](/D/dist-marg) of $Y$ using the [law of conditional probability](/D/prob-cond).

D/dist-joint.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, a joint distribution of $X$ and $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ and $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$.
+**Definition:** Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, a joint distribution of $X$ and $Y$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ and $Y = y$ for each possible combination of $x \in \mathcal{X}$ and $y \in \mathcal{Y}$.
 
 * The joint distribution of two scalar [random variables](/D/rvar) is called a bivariate distribution.
 

D/dist-marg.md

@@ -27,4 +27,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the marginal distribution of $X$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ irrespective of the value of $Y$ for each possible value $x \in \mathcal{X}$. The marginal distribution can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ using the [law of marginal probability](/D/prob-marg).
+**Definition:** Let $X$ and $Y$ be [random variables](/D/rvar) with sets of possible outcomes $\mathcal{X}$ and $\mathcal{Y}$. Then, the marginal distribution of $X$ is a [probability distribution](/D/dist) that specifies the probability of the event that $X = x$ irrespective of the value of $Y$ for each possible value $x \in \mathcal{X}$. The marginal distribution can be obtained from the [joint distribution](/D/dist-joint) of $X$ and $Y$ using the [law of marginal probability](/D/prob-marg).

D/dist.md

@@ -27,4 +27,4 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar) with the set of possible outcomes $\mathcal{X}$. Then, a probability distribution of $X$ is a mathematical function that gives the [probabilities](/D/prob) of occurrence of all possible outcomes $x \in \mathcal{X}$ of this random variable.
+**Definition:** Let $X$ be a [random variable](/D/rvar) with the set of possible outcomes $\mathcal{X}$. Then, a probability distribution of $X$ is a mathematical function that gives the [probabilities](/D/prob) of occurrence of all possible outcomes $x \in \mathcal{X}$ of this random variable.

D/exp.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to be exponentially distributed with rate (or, inverse scale) $\lambda$
+**Definition:** Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to be exponentially distributed with rate (or, inverse scale) $\lambda$
 
 $$ \label{eq:exp}
 X \sim \mathrm{Exp}(\lambda) \; ,

D/gam.md

@@ -28,7 +28,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a gamma distribution with shape $a$ and rate $b$
+**Definition:** Let $X$ be a [random variable](/D/rvar). Then, $X$ is said to follow a gamma distribution with shape $a$ and rate $b$
 
 $$ \label{eq:gam}
 X \sim \mathrm{Gam}(a, b) \; ,

D/matn.md

@@ -27,7 +27,7 @@ username: "JoramSoch"
 ---
 
 
-**Definition**: Let $X$ be an $n \times p$ [random matrix](/D/rmat). Then, $X$ is said to be matrix-normally distributed with mean $M$, [covariance](/D/covmat) across rows $U$ and [covariance](/D/covmat) across columns $V$
+**Definition:** Let $X$ be an $n \times p$ [random matrix](/D/rmat). Then, $X$ is said to be matrix-normally distributed with mean $M$, [covariance](/D/covmat) across rows $U$ and [covariance](/D/covmat) across columns $V$
 
 $$ \label{eq:matn}
 X \sim \mathcal{MN}(M, U, V) \; ,