## Encyclopedia > Chi-square distribution

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# Chi-square distribution

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

$\chi^2 = Z_1^2 + \cdots + Z_k^2$
where Z1, ..., Zk are independent normal variables, 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.

$p_k(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)} x^{k/2 - 1} e^{-x/2} \quad \mbox{ for }x > 0$ 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.

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