# 4.7: Chi-Squared Distributions

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.

1. The mgf of $$X$$ is given by
$$M_X(t) = \frac{1}{(1-2t)^{k/2}},\quad \text{for}\ t<\frac{1}{2}\notag$$
2. The mean of $$X$$ is $$\text{E}[X] = k$$, i.e., the degrees of freedom.
3. 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

1. 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.
2. 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.
3. If $$X\sim\chi^2(k_1)$$ and $$Y\sim\chi^2(k_2)$$ are independent random variables, then $$X+Y\sim\chi^2(k_1+k_2)$$.