Section 9.3 Chi-Square Distribution
The following distribution is related to both the Normal Distribution and to the Gamma Distribution. Initially, consider a gamma distribution with probability function
\begin{equation*}
\frac{x^{r-1} \cdot e^{-x / \mu}}{\Gamma(r) \cdot \mu^r}.
\end{equation*}
Replacing \(\mu = 2\) and r with r/2 gives
\begin{equation*}
\frac{x^{r/2-1} \cdot e^{-x/2}}{\Gamma(r/2) \cdot 2^{r/2}}
\end{equation*}
which is given a special name below.
Definition 9.3.1 Chi-Square Probability Function
Given an natural number r, suppose X is a random variable over the space \(R = (0,\infty)\) with probability function given by
\begin{equation*}
f(x) = \frac{x^{r/2-1} e^{-x/2} }{\Gamma(r/2) 2^{r/2}}.
\end{equation*}
Then X has a Chi-Square distribution with r degrees of freedom. This is often denoted \(\chi^2(r)\text{.}\)
Theorem 9.3.2 \(\chi^2\) statistics
\begin{equation*}
\mu = r
\end{equation*}
\begin{equation*}
\sigma^2 = 2r
\end{equation*}
\begin{equation*}
\gamma_1 = 2 \sqrt{2/r}
\end{equation*}
\begin{equation*}
\gamma_2 = \frac{12}{r} + 3
\end{equation*}
Theorem 9.3.3 Relationship between Normal and \(\chi^2\)
If \(Z_1, Z_2, ..., Z_r\) are r standard normal variables, then
\begin{equation*}
X = \sum_{k=1}^r Z_k^2
\end{equation*}
is \(\chi^2(r)\text{.}\) It also can be difficult to compute Chi-Square probabilities manually so you will perhaps want to use a numerical approximation in this case as well. The TI graphing calculator can be used with \(\chi ^2\)cdf(a,b,r). Or, you can use the calculator below.