Chi-Square Distribution

Section 9.3 Chi-Square Distribution

The following distribution is related to both the Normal Distribution and to the Gamma Distribution 8.4.3. Initially, consider a gamma distribution with probability function given by the formula

$\begin{equation*} y = \frac{x^{r-1} \cdot e^{-x / \mu}}{\Gamma(r) \cdot \mu^r}. \end{equation*}$

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Replacing

$\mu = 2$ and r with r/2 gives the special case

$\begin{equation*} y = \frac{x^{r/2-1} \cdot e^{-x/2}}{\Gamma(r/2) \cdot 2^{r/2}} \end{equation*}$

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which is given a special name below.

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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,0,40).show(title="Chi-Square Graph",figsize=[5,3])

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As with all distributions before, one can determine the mean, variance, skewness and kurtosis for a general

$\chi^2$ distribution directly. However, one can also note that these should be special cases of the gamma distribution and indeed that is the case.

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Theorem 9.3.2. $\chi^2$ statistics.

$\begin{equation*} \mu = r \end{equation*}$

$\begin{equation*} \sigma^2 = 2r \end{equation*}$

$\begin{equation*} \gamma_1 = 2 \sqrt{2/r} \end{equation*}$

$\begin{equation*} \gamma_2 = \frac{12}{r} + 3 \end{equation*}$

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Proof.

Consider the formulas developed for the Gamma distribution 8.4.7 with that $\mu=2$ amd $r$ replaced with $r/2\text{.}$ This gives

\begin{equation*} \mu = \frac{r}{2}2 = r \end{equation*}

\begin{equation*} \sigma^2 = \frac{r}{2}2^2 = 2r \end{equation*}

\begin{equation*} \gamma_1 = \frac{2}{\sqrt{\frac{r}{2}}} = \frac{2^{3/2}}{\sqrt{r}} \end{equation*}

\begin{equation*} \gamma_2 = \frac{6}{\frac{r}{2}} + 3 = \frac{12}{r} + 3 \end{equation*}

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So, if the

$\chi^2$ distribution is based upon the gamma distribution, why might one want to save it for this point in time? The Gamma distribution has a specific setup for the random variable for solving a particular problem...finding the probability that it takes an amount of time in order to reach a defined number of successes.

$\chi^2$ simply is created by using a redesign of the gamma formula but with no particular problem to solve in mind. However,

$\chi^2$ has a number of properties that are useful for making inferences from sample data as you will see later. The theorem below shows an important relationship between the

$\chi^2$ distribution and the standard normal distribution.

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To express this relationship requires the use of multi-variate statistics. This text is focused on single-variable statistics so what follows will be a little careless. Take a followup course to rigorously develop what to do with various functions on random variables. For now, just consider this: Suppose that you plan on doing an experiment on some distribution with a given mean and given variance and that that experiment has random variable

$X_1\text{.}$ Planning to do the experiment again results in a random variable

$X_2\text{.}$ Continuing, you will get (say) n random experiments planned that will result in n different random variables

$\begin{equation*} X_1, X_2, X_3, ... , X_n\text{.} \end{equation*}$

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You can then create a new random variable that might be a combination of those variables. One such random variable that is often chosen is by taking the sum of these theoretical values such as

$\begin{equation*} Y = \sum_{k=1}^n X_k = X_1 + X_2 + X_3 + ... + X_n \end{equation*}$

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or the average of these variables to create the variable

$\overline{X}$ where

$\begin{equation*} \overline{X} = \frac{\sum_{k=1}^n X_k}{n} = \frac{X_1 + X_2 + X_3 + ... + X_n}{n}. \end{equation*}$

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Below, notice that the new variable created comes by taking the sum of the squares of standard normal variables. This is indeed yet another possible function on random variables that establishes a relationship between normal and

$\chi^2\text{.}$

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

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

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

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

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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

$P(a \le X \le b) \approx \chi^2$ cdf(a,b,r). Or you can use the interactive cell below.

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# Chi-Square Calculator
pretty_print("Calculator for Chi Square Probabilities")
@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]
#    T = RealDistribution('chisquared', r)   # use built-in 
#    P = T.cum_distribution_function(b)-T.cum_distribution_function(a)
​
    pretty_print(html("$$ P("+str(a)+" < X < "
          +str(b)+") \\approx "+str(P)+"$$"))

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Checkpoint 9.3.4.

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As with the normal distribution, there is also a way to compute the inverse Chi-Square function using Sage.

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# Inverse Chi-Square Calculator
pretty_print("Calculator for Inverse Chi-Square")
var("x")
@interact(layout=dict(top=[['p0', 'r0']]))
def _(p0=input_box(1/2,width=10,label='probability= '),
      r0=input_box(5,width=8,label='$$r= $$')):
    T = RealDistribution('chisquared', r0)   # use built-in
    x0 = T.cum_distribution_function_inv(p0)  # rel tol 1e-14
    pretty_print(html("$$ P(X < x_0) = "+str(p0)+" \\; \\; " +
                      "\\Rightarrow \\; \\; x_0 ="+str(x0)+"$$"))
    f = x^(r0/2-1)*e^(-x/2)/(gamma(r0/2)*2^(r0/2))
    G = (plot(f,(x,0,x0),fill=True, fillcolor='green') +
          plot(f,(x,0,4*r0),thickness=3, color='black'))
    G.show(figsize=[5,3])

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Essentials of Mathematical Probability and Statistics

Section 9.3 Chi-Square Distribution

Definition 9.3.1. Chi-Square Probability Function.

Theorem 9.3.2. $\chi^2$ statistics.

Proof.

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

Checkpoint 9.3.4.

Checkpoint 9.3.5.

Section 9.3 Chi-Square Distribution

Definition 9.3.1. Chi-Square Probability Function.

Theorem 9.3.2. χ2\chi^2 statistics.

Proof.

Theorem 9.3.3. Relationship between Normal and χ2\chi^2.

Checkpoint 9.3.4.

Checkpoint 9.3.5.

Theorem 9.3.2. $\chi^2$ statistics.

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