Chi-square distribution
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 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.
Its probability density function is
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.
Referenced By
ANOVA | ANOVA/DegreesOfFreedom | ANOVA/Fixed | ANOVA/Random | Analysis of Variance | Analysis of variance/Degrees of freedom | Analysis of variance/Fixed effects model | Analysis of variance/Random effects models | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | List of probability topics | List of statistical topics | Non-parametric | Non-parametric statistic | Non-parametric statistics | ProbabilityApplications | Probability Applications
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