Distribusyong hiperheometriko

Sa teoriya ng probabilidad at estadistika, ang distribusyong hiperheometriko ay isang diskretong distribusyong probabilidad na naglalarawan ng probabilidad ng $k$ mga tagumpay sa $n$ mga paghugot mula sa may hangganang populasyon ng sukat $N$ na naglalaman ng $m$ mga tagumpay nang walang pagpapalit.

Hypergeometric

Parameters	${\begin{aligned}N&\in \left\{1,2,\dots \right\}\\m&\in \left\{0,1,2,\dots ,N\right\}\\n&\in \left\{1,2,\dots ,N\right\}\end{aligned}}\,$
Support	$\scriptstyle {k\,\in \,\left\{\max {(0,\,n+m-N)},\,\dots ,\,\min {(m,\,n)}\right\}}\,$
PMF	${{{m \choose k}{{N-m} \choose {n-k}}} \over {N \choose n}}$
CDF	$1-{{{n \choose {k+1}}{{N-n} \choose {m-k-1}}} \over {N \choose m}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-m,\ k+1-n\\k+2,\ N+k+2-m-n\end{array}};1\right]$
Mean	$n{m \over N}$
Mode	$\left\lfloor {\frac {(n+1)(m+1)}{N+2}}\right\rfloor$
Variance	$n{m \over N}{(N-m) \over N}{N-n \over N-1}$
Skewness	${\frac {(N-2m)(N-1)^{\frac {1}{2}}(N-2n)}{[nm(N-m)(N-n)]^{\frac {1}{2}}(N-2)}}$
Ex. kurtosis	$\left.{\frac {1}{nm(N-m)(N-n)(N-2)(N-3)}}\cdot \right.$ ${\Big [}(N-1)N^{2}{\Big (}N(N+1)-6m(N-m)-6n(N-n){\Big )}+$ $6nm(N-m)(N-n)(5N-6){\Big ]}$
MGF	${\frac {{N-m \choose n}\scriptstyle {\,_{2}F_{1}(-n,-m;N-m-n+1;e^{t})}}{N \choose n}}\,\!$
CF	${\frac {{N-m \choose n}\scriptstyle {\,_{2}F_{1}(-n,-m;N-m-n+1;e^{it})}}{N \choose n}}$

Tignan din

• Distribusyong binomial