For any positive integer , the **chi-square distribution** with *k* degrees of freedom is the probability distribution of the random variable

*Z*

_{1}, ...,

*Z*

_{k}are independent normal variabless, each having expected value 0 and variance 1.

The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two chi-squared random variables.

Its probability density function is

*p*

_{k}(

*x*) = 0 for

*x*≤0. Here Γ denotes the Gamma function.

The expected value of a random variable having chi-square distribution with *k* degrees of freedom is *k* and the variance is 2*k*. Note that 2 degrees of freedom leads to an exponential distribution.

The chi-square distribution is a special case of the gamma distribution.