Chi-Square Distribution

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Suppose we have n independently and identically distributed standard normal random variables for . where n is a positive integer. The distribution of the sum of the squared is referred to as the Chi-square distribution, with parameter df, or written shortly as . The parameter represents the degrees of freedom of the distribution, with . The expected value and variance of Chi-square distribution are given as This diagram plots the density functions for some Chi-square distributions, with different values for the degrees of freedom .

Nl s2 29a 1.gif

The Chi-square, t-, and F- distributions are distributions that are functions of Normal random variables that are particularly useful in statistics. On the chi-square distribution. The parameter denotes the degrees of freedom. The degrees of freedom reflects the number of independent random variables included in the sum Y. If the random variables are independent from each other, then squaring and summing them does not change their properties. In this example, the random variable : will have the Chi-square distribution with degrees of freedom. The shape of the density function will depend on the parameter . For  and , the distribution follows a monoton structures. For small values of , the -distribution will be skewed to the right. However, as increases the -distribution will tend towards the normal density function. The -distribution is tabulated for a number of values of .