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.
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
