Hypergeometric Distribution (Hypergeom₍N,K,n₎) ○◂｜_append｜___

Hypergeometric	…	…	…
Probability mass function	…	…	…
Cumulative distribution function	…	…	…
Notation	$Hypergeometric (N, K, n)$	…	…
Parameters	$N K n \in {0, 1, 2, \dots} \in {0, 1, 2, \dots, N} \in {0, 1, 2, \dots, N}$	…	…
Support	$k \in {m a x (0, n + K - N), \dots, m i n (n, K)}$	…	…
PMF	$\frac{( k K ) ( n - k N - K )}{( n N )}$	…	…
CDF	$1 - \frac{( k + 1 n ) ( K - k - 1 N - n )}{( K N )}_{3} F_{2} [1, k + 1 - K, k + 1 - n k + 2, N + k + 2 - K - n; 1],$ where $_{p} F_{q}$ is the generalized hypergeometric function	…	…
Mean	$n \frac{K}{N}$	…	…
Median	$⌊ n \frac{K}{N} ⌋$ or $⌈ n \frac{K}{N} ⌉$ ¹	…	…
Mode	$⌈ \frac{( n + 1 ) ( K + 1 )}{N + 2} ⌉ - 1, ⌊ \frac{( n + 1 ) ( K + 1 )}{N + 2} ⌋$	…	…
Variance	$n \frac{K}{N} \frac{N - K}{N} \frac{N - n}{N - 1}$	…	…
Skewness	$\frac{( N - 2 K ) ( N - 1 ) ^{\frac{1}{2}} ( N - 2 n )}{[ n K ( N - K ) ( N - n ) ] ^{\frac{1}{2}} ( N - 2 )}$	…	…
Excess kurtosis	$\frac{1}{n K ( N - K ) ( N - n ) ( N - 2 ) ( N - 3 )} \cdot$ $[(N - 1) N^{2} (N (N + 1) - 6 K (N - K) - 6 n (N - n))$ $+ 6 n K (N - K) (N - n) (5 N - 6)]$	…	…
MGF	$\frac{( n N - K ) _{2} F _{1} ( - n , - K ; N - K - n + 1 ; e ^{t} )}{( n N )}$	…	…
CF	$\frac{( n N - K ) _{2} F _{1} ( - n , - K ; N - K - n + 1 ; e ^{i t} )}{( n N )}$	…	…

(echo:: @ ᯤ)

Cooper, Joshua N; Ellis, Robert B (2009). “Linearly bounded liars, adaptive covering codes, and deterministic random walks”. arXiv preprint arXiv:0909.0029., p.15. ↩

⏾ Concept Map