4.7: Chi-Squared Distributions

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In this section, we introduce the chi-squared distributions, which are very useful in statistics.

Chi-Squared Distributions

Definition $\PageIndex{1}$

A random variable $X$ has a chi-squared distribution with $k$ degrees of freedom, where $k$ is a positive integer, write $X\sim\chi^2(k)$, if $X$ has pdf given by
$$f(x) = \left\{\begin{array}{l l}
\displaystyle{\frac{1}{\Gamma(k/2)2^{k/2}} x^{k/2-1} e^{-x/2}}, & \text{for}\ x\geq 0, \\
0 & \text{otherwise,}
\end{array}\right. \notag$$

Figure 1: Graph of pdf for $\chi^2(1)$ distribution.

The chi-squared distributions are a special case of the gamma distributions with $\alpha = \frac{k}{2}, \lambda=\frac{1}{2}$, which can be used to establish the following properties of the chi-squared distribution.

Properties of Chi-Squared Distributions

If $X\sim\chi^2(k)$, then $X$ has the following properties.

The mgf of $X$ is given by
$$M_X(t) = \frac{1}{(1-2t)^{k/2}},\quad \text{for}\ t<\frac{1}{2}\notag$$
The mean of $X$ is $\text{E}[X] = k$, i.e., the degrees of freedom.
The variance of $X$ is $\text{Var}(X) = 2k$, i.e., twice the degrees of freedom.

Note that there is no closed form equation for the cdf of a chi-squared distribution in general. But most graphing calculators have a built-in function to compute chi-squared probabilities. On the TI-84 or 89, this function is named "$\chi^2$cdf''.

The main applications of the chi-squared distributions relate to their importance in the field of statistics, which result from the following relationships between the chi-squared distributions and the normal distributions.

Relationships of Chi-Squared Distributions

If $Z$ is a standard normal random variable, i.e., $Z\sim N(0,1)$, then the distribution of $Z^2$ is chi-squared with $k=1$ degree of freedom.
If $X_1, \ldots, X_n$ is a collection of independent, chi-squared random variables each with 1 degree of freedom, i.e., $X_i\sim\chi^2(1)$, for each $i=1,\ldots,n$, then the sum $X_1+\cdots+X_n$ is also chi-squared with $k=n$ degrees of freedom.
If $X\sim\chi^2(k_1)$ and $Y\sim\chi^2(k_2)$ are independent random variables, then $X+Y\sim\chi^2(k_1+k_2)$.

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Definition \(\PageIndex{1}\)

Properties of Chi-Squared Distributions

Relationships of Chi-Squared Distributions