Negative hypergeometric distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ negative hypergeometric (n_{1}, n_{2}, n_{3}) is used to indicate that the random variable X has the negative hypergeometric distribution with parameters n_{1}, n_{2}, and n_{3}. A negative hypergeometric random variable X with parameters n_{1}, n_{2}, and n_{3} has probability mass function

f(x) = \begin{matrix}\binom{n_{1}+x−1}{x} & \binom{n_{3}−n_{1}+n_{2}−x−1}{n_{2}−x} \\ ( & {n_{3}+n_{2}−1 \atop n_{2}} & ) \end{matrix} x = max\{0,n_{1}+ n_{2}− n_{3}\} ,...,n_{2}

for all n_{1} ∈ {1,2,...,n_{2}}, n_{2},n_{3} ∈ {1,2,...}. The probability mass function for n_{1} = 5,n_{2} = 20, and n_{3} = 30 is illustrated below.

PSfrag replacements

x
f(x)

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, and variance of X are

E[X] = \frac{n_{1}n_{2}}{n_{3}} V [X] = \frac{n_{1}n_{2}(n_{3}+ n_{2})}{n_{3}(n_{3}+ 1)} ( 1 − \frac{n_{1}}{n_{3}})

The skewness, and kurtosis of X are mathematically intractable.

APPL verification: The APPL statements

assume(0 ¡ n1 ¡ n2); X := [[x -¿ binomial(n1 + x - 1, x) * binomial(n3 - n1 + n2 - x - 1, n2 - x)

/ (binomial(n3 + n2 - 1, n2))], [0 .. n2], [”Discrete”,”PDF”]];

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

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