# Hypergeometric distribution Measures

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

## 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.

`\text{mode}(X) = |__A__|`

If A is integer then there are two modes,
`\text{mode}(X) = A-1 \text{ and } A`

## See also

Hypergeometric distribution probabilities

Inverse hypergeometric distribution

Hypergeometric distribution Histogram

Statistics Calculators