Noncentral t distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ noncentral t (n, δ) is used to indicate that the random variable X has the noncentral t distribution with positive integer parameter n and real-valued noncentrality parameter δ. A noncentral t random variable X with parameters n and δ has probability density function

f(x) = \frac{n^{n/2}e^{−δ^{2}/2}}{\sqrt{π}Γ(n/2)(n + x^{2})^{(n+1)/2}} · ∑_{i=0}^{∞} \frac{Γ[(n + i + 1)/2]}{i!} ( \frac{xδ\sqrt{2}}{\sqrt{n + x^{2}}} ) ^{i} − ∞ < x < ∞,

for all δ and any positive integer n.

The cumulative distribution, survivor, hazard, cumulative hazard, inverse distribution, moment generating, and characteristic functions on the support of X are mathematically intractable.

The population mean and variance of X are

E[X] = δ\sqrt{\frac{n}{2}} \frac{Γ((n − 1)/2)}{Γ(n/2)} n > 1

V [X] = \frac{n(1 + δ^{2})}{n − 2} − \frac{δ^{2}n}{2}( \frac{Γ((n − 1)/2)}{Γ(n/2)} ) ^{2} n > 2.