Let X be a normal random variable with mean δ and variance 1 and S2 be a chisquare random variable with degrees of freedom (df) n. If X and S2 are independent, then the distribution of the ratio

√ nX/S is called the noncentral t distribution with

the degrees of freedom n and the noncentrality parameter δ. The probability density function is given by

f(x|n, δ) = n n/2 exp(−δ2/2)√

π Γ(n/2)(n+ x2)(n+1)/2

Γ[(n+ i+ 1)/2]