Chi distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ χ(n) is used to indicate that the random variable X has the χ (chi) distribution with n degrees of freedom, where n is a positive integer. A χ random variable X with n degrees of freedom has probability density function

f(x) = \frac{1}{2^{n/2−1} Γ(n/2)}x^{n−1} e^{−x^{2}/2} x > 0.

The probability density function, where n = 1, 2, and8 is illustrated below.

PSfrag replacements

n = 1
n = 2
n = 8
x
f(x)

The cumulative distribution function of X can be written as an integral that can’t be evaluated in closed form. The expected value of X^{k} is

E [X^{k}]= \frac{2^{k/2}Γ(k/2 + n/2)}{Γ(n/2)}

for k = 1, 2, .... Using this result, the mean and variance are

E[X] = \frac{\sqrt{2}Γ(1/2 + n/2)}{Γ(n/2)} V [X] = n − μ^{2}

because

E [X^{2}]= \frac{2Γ(n/2 + 1)}{Γ(n/2)} = n.

The skewness and kurtosis are also written in terms of the gamma function. The mode is \sqrt{n − 1} .

APPL verification: The APPL statements

X := ChiRV(n); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the population mean, variance, skewness, kurtosis, and moment generating function.