Chi-Square Distribution

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

$\chi^2(r)\text{.}$

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# Chi-Square Grapher
@interact
def _(r=slider(1,20,1,3,label='r =')):
    f = x^(r/2-1)*e^(-x/2)/(gamma(r/2)*2^(r/2))
    plot(f,x,0,20).show()

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 $\chi^2$

$Z_1, Z_2, ..., Z_r$ are r standard normal variables, then

$\begin{equation*} X = \sum_{k=1}^r Z_k^2 \end{equation*}$

$\chi^2(r)\text{.}$

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.

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# Chi-Square Calculator
@interact(layout=dict(top=[['a', 'b']],bottom=[['r']]))
def _(a=input_box(0,width=10,label='a = '),b=input_box(2,width=10,label='b = '),r=input_box(2,width=8,label='r =')):
    f = x^(r/2-1)*e^(-x/2)/(gamma(r/2)*2^(r/2))
    P = numerical_integral(f,a,b)[0]
    print "P("+str(a)+" < X < "+str(b)+") ~= "+str(P)

Section 9.3 Chi-Square Distribution

Definition 9.3.1 Chi-Square Probability Function

Theorem 9.3.2 χ2\chi^2 statistics

Theorem 9.3.3 Relationship between Normal and χ2\chi^2

Theorem 9.3.2 $\chi^2$ statistics

Theorem 9.3.3 Relationship between Normal and $\chi^2$