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

f(x) = \frac{\binom{n}{x} ∏_{j=0}^{x−1}(p + jβ) ∏_{k=0}^{n−x−1} (1 − p + kβ)}{∏_{i=0}^{n−1}(1 + iβ)} x = 0,1,2,..., n,

for all n = 1,2,..., 0 < p < 1, and β > 0. The cumulative distribution, survivor function, hazard function, cumulative hazard function, and inverse distribution function, moment generating function, and characteristic function on the support of X are mathematically intractable. The population mean of X is

E[X] = −\frac{sin( \frac{π(−1+β+p)}{β} ) pnsin( \frac{π(nβ+1)}{β} )}{sin( \frac{π}{β}) sin( \frac{π(nβ−β−p+1)}{β} )} .

APPL verification: The APPL statements

X := [[x -¿ binomial(n, x) * product(p + j * beta, j = 0 .. x - 1) * product(1 - p + k * beta, k = 0 .. n - x - 1) / (product(1 + i * beta, i = 0 .. n - 1))], [0 .. n], [”Discrete”, ”PDF”]];

Mean(X); Variance(X); Skewness(X); MGF(X);

return the population mean, variance, skewness, and moment generating function.