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*}

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*}

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

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.

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*}

Proof.

Consider the formulas developed for the Gamma distribution 8.4.8 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*}

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.

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*}

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*}

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*}

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

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