3.10.3 Chi-squared Distributions

3.10.3  Chi-squared Distributions

Suppose Z is a standard normal random variable. How is Z 2 distributed? The answer is a chi-squared distribution. More generally, let Z1, Z2, … Zν be ν independent standard normal random variables, and let δ1, δ2, … δν be constants. Then the random variable


has a chi-squared distribution with ν degrees of freedom and noncentrality parameter7


We denote a chi-squared distribution χ2(ν,δ2). If δ2 = 0, the distribution is said to be centrally chi-squared. Otherwise, it is said to be noncentrally chi-squared. The PDF for a central chi-squared distribution is


For noncentral chi-squared distributions, this generalizes to


where Γ( ) denotes the gamma function8


The PDF of a chi-squared distribution is illustrated in Exhibit 3.19.

Exhibit 3.19: PDF of a chi-squared distribution.

The expectation, standard deviation, skewness, and kurtosis of a chi-squared distribution are