# Hypergeometric distribution Measures

Calculates probability mass function (PMF), mean, variance, mode, standard deviation, kurtosis and skewness of an hypergeometric distribution.
n <= M
N <= M

## Notations

• X : a random variable following an hypergeometric distribution: X ~ N(M, n, N)
• M : population size
• n : number of successes in population
• N : number of draws
• P(X = k) : probability to have exactly k successful draws
• ([n], [k]) : binomial coefficient n choose k

([n], [k]) = (n!)/(k! * (n-k)!)

## Probability Mass Function (PMF)

P(X = k) = (([n], [k]) * ([M-n], [N-k]))/(([M], [N]))

## Mean (or Expected value)

E(X) = Nn/M

## Standard deviation

sigma(X) = sqrt(Nn/M((M-n)/M)((M-N)/(M-1)))

## Variance

V(X) = Nn/M((M-n)/M)((M-N)/(M-1))

## Skewness

S(X) = ((M-2*n)*sqrt(M-1)*(M-2*N))/(sqrt(N*n*(M-n)*(M-N))*(M-2))

## Kurtosis

K(X) = 1/(N*n*(M-n)*(M-N)*(M-2)*(M-3)) . Q Q = (M-1)*M^2*(M*(M+1)-6*n*(M-n)-6*N*(M-N))+6*N*n*(M-n)*(M-N)*(5*M-6)

Q has no particular signification. It is used only to simplify the formula.

## Mode

We denote A = ((N+1)*(n+1))/(M+2)
|__A__| : integer part of A.

If A is not integer (this is generally the case),
\text{mode}(X) = |__A__|
If A is integer then there are two modes,
\text{mode}(X) = A-1 \text{ and } A