Let X1, . . . , Xn be independent standard normal random variables. The distribution

of X = n∑ i=1

X2i is called the chi-square distribution with degrees of freedom (df) n,

and its probability density function is given by

f(x|n) = 1 2n/2Γ(n/2)

e−x/2xn/2−1, x > 0, n > 0. (12.1)

The chi-square random variable with df = n is denoted by χ2n. Since the probability density function is valid for any n > 0, alternatively, we can define the chi-square distribution as the one with the probability density function (12.1). This latter definition holds for any n > 0. The cdf is given by

F (x|n) = 1 2n/2Γ(n/2)

e−t/2tn/2−1dt, n > 0.