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    Hypergeometric Distribution

    Statistical properties

    mean | (m n)/N standard deviation | sqrt((m n (N - m) (N - n))/(N - 1))/N variance | (m n (1 - m/N) (N - n))/((N - 1) N) skewness | (sqrt(N - 1) (N - 2 m) (N - 2 n))/((N - 2) sqrt(m n (N - m) (N - n))) kurtosis | ((N - 1) N^2 ((3 m (N - m) (n^2 (-N) + (n - 2) N^2 + 6 n (N - n)))/N^2 - 6 n (N - n) + N (N + 1)))/(m n (N - 3) (N - 2) (N - m) (N - n))

    Probability density function (PDF)

    P (X = x) = piecewise | (binomial(m, x) binomial(N - m, n - x))/binomial(N, n) | 0<=x<=m ∧ m + n - N<=x<=m ∧ 0<=x<=n ∧ m + n - N<=x<=n 0 | (otherwise)

    Plots of PDF for typical parameters

    Plots of PDF for typical parameters

    Cumulative distribution function (CDF)

    P (X<=x) = piecewise | 1 - (m! (N - m)! _3 F^~_2(1, -n + floor(x) + 1, -m + floor(x) + 1;floor(x) + 2, -m - n + N + floor(x) + 2;1))/(binomial(N, n) (-floor(x) + m - 1)! (-floor(x) + n - 1)!) | 0<=x ∧ m + n - N<=x ∧ x<m ∧ x<n 1 | x>=m ∨ x>=n

    Plots of CDF for typical parameters

    Plots of CDF for typical parameters

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