Chi-Square Distribution

The chi-square distribution is given by,

\[ \frac{\frac{1}{2}^{n/2}}{\Gamma(n/2)} x^{\frac{n}{2} - 1}e^{- \frac{1}{2}x} \]

Member of the Gamma Distribution

The chi-square distribution is a member of the gamma distribution,

\[ \chi^n_k = \text{gamma}(n/2, 1/2) \]

Relation to Normal Distribution

The \(\chi^2_n\) distribution with \(n\) degrees of freedom is the sum of squares of \(n\) iid standard normal random variables \(Z_1, Z_2, \ldots, Z_n\),

\[ \chi^2_n = \sum_k^n Z_k^2 \]

Motivating Example - The Squared Standard Normal : Consider the standard normal random variable \(Z\). We can determined its PDF \(X=Z^2\) by properties of two-to-one function transformation \(g(x)\):

$$
\begin{gather*}
g(x) = z^2\\
g'(x) = 2z
\end{gather*}
$$

$$
\begin{align*}
f_X &= \frac{f_Z(\sqrt{x})}{g'(\sqrt{x})} + \frac{f_Z(-\sqrt{x})}{\abs{g'(-\sqrt{x})}}\\
&= \frac{1}{\sqrt{2\pi^2}}\left[\frac{e^{-\frac{1}{2}x}}{2\sqrt{x}} + \frac{e^{-\frac{1}{2}x}}{2\sqrt{x}}\right]\\
&= \frac{1}{\sqrt{2\pi^2}}x^{-\frac{1}{2}}e^{-\frac{1}{2}x}\\
f_X &= \frac{\tfrac{1}{2}^\tfrac{1}{2}}{\pi}x^{\frac{1}{2}-1}e^{-\frac{1}{2}x}
\end{align*}
$$

We now see that $Z^2 = \chi_1^2$ the squared standard normal is the chi-square with one degree of freedom.

Unbiased Estimator of SD with Known Mean

Given iid \(\text{Normal}(\mu, \sigma^2)\) random variables \(X_1, X_2, \ldots, X_n\), the unbiased estimator of \(\sigma\) is,

\[\begin{split} \begin{gather*} Y = \frac{\sigma^2}{n}\sum_{i=1}^n\\ E(Y) = \sigma^2 \end{gather*} \end{split}\]