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

f(x) = ∑_{i=0}^{∞} \frac{Γ(i + β + γ)}{Γ(γ)Γ(i + β)} ( \frac{e^{−δ/2}}{i!} ) ( \frac{δ}{2}) ^{i}x^{i+β−1} (1 − x)^{γ−1} 0 < x < 1,

for all β > 0, γ > 0, δ > 0.

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, variance, skewness, and kurtosis of X are mathematically intractable.