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

where Z1, ..., Zk 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.

and pk(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 2k. Note that 2 degrees of freedom leads to an exponential distribution.

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