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

f(x) = \frac{x^{β−1} (x + 1)^{−β−γ}}{B(β,γ)} x > 0,

for β ≥ 1 and γ ≥ 1. The probability density function with four different parameter settings is illustrated below.

PSfrag replacements β = 1

γ = 1
γ = 2
β = 2

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 of X is

E[X] = \frac{Γ(−1 + γ) Γ(1 + β)}{Γ(β + γ) B(β,y)} .

APPL verification: The APPL statements

assume(beta ¿ 1); assume(y ¿ 1); X := [[x -¿ x ˆ (beta - 1) * (1 + x) ˆ (-beta - y) / Beta(beta, y)],

[0, infinity],[”Continuous”, ”PDF”]]; Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the population mean, variance, skewness, and kurtosis.