Chi-square distribution

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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, ..., Z_k are independent normal variabless, each having expected value 0 and variance 1. This distribution is usually written

If independent linear homogeneous constraints are imposed on these variables, the distribution of conditional on these constriants is , justifying the term "degrees of freedom". The characteristic function of the Chi-square distribution is

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

and p_k(x) = 0 for x≤0. Here Γ denotes the gamma function.

The normal approximation

If , then as tends to infinity, the distribution of tends to normality. However, the tendency is slow (the skewness is and the kurtosis is ) and two transformations are commonly considered, each of which approaches normality faster than itself:

Fisher showed that is approximately normally distributed with mean and unit variance.

Wilson and Hilferty showed in 1931 that is approximately normally distributed with mean and variance .

The expected value of a random variable having chi-square distribution with k degrees of freedom is k and the variance is 2k. The median is given approximately by

Note that 2 degrees of freedom leads to an exponential distribution.

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

See Cochran's theorem.