Hypergeometrinen jakauma
Todennäköisyysfunktio
Kertymäfunktio
Parametrit
N
∈
{
0
,
1
,
2
,
…
}
K
∈
{
0
,
1
,
2
,
…
,
N
}
n
∈
{
0
,
1
,
2
,
…
,
N
}
{\displaystyle {\begin{aligned}N&\in \left\{0,1,2,\dots \right\}\\K&\in \left\{0,1,2,\dots ,N\right\}\\n&\in \left\{0,1,2,\dots ,N\right\}\end{aligned}}\,}
Määrittelyjoukko
k
∈
{
max
(
0
,
n
+
K
−
N
)
,
…
,
min
(
K
,
n
)
}
{\displaystyle \scriptstyle {k\,\in \,\left\{\max {(0,\,n+K-N)},\,\dots ,\,\min {(K,\,n)}\right\}}\,}
Pistetodennäköisyysfunktio
(
K
k
)
(
N
−
K
n
−
k
)
(
N
n
)
{\displaystyle {{{K \choose k}{{N-K} \choose {n-k}}} \over {N \choose n}}}
Kertymäfunktio
1
−
(
n
k
+
1
)
(
N
−
n
K
−
k
−
1
)
(
N
K
)
3
F
2
[
1
,
k
+
1
−
K
,
k
+
1
−
n
k
+
2
,
N
+
k
+
2
−
K
−
n
;
1
]
{\displaystyle 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right]}
Odotusarvo
n
K
N
{\displaystyle n{K \over N}}
Moodi
⌊
(
n
+
1
)
(
K
+
1
)
N
+
2
⌋
{\displaystyle \left\lfloor {\frac {(n+1)(K+1)}{N+2}}\right\rfloor }
Varianssi
n
K
N
(
N
−
K
)
N
N
−
n
N
−
1
{\displaystyle n{K \over N}{(N-K) \over N}{N-n \over N-1}}
Vinous
(
N
−
2
K
)
(
N
−
1
)
1
2
(
N
−
2
n
)
[
n
K
(
N
−
K
)
(
N
−
n
)
]
1
2
(
N
−
2
)
{\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}}
Huipukkuus
1
n
K
(
N
−
K
)
(
N
−
n
)
(
N
−
2
)
(
N
−
3
)
⋅
{\displaystyle \left.{\frac {1}{nK(N-K)(N-n)(N-2)(N-3)}}\cdot \right.}
[
(
N
−
1
)
N
2
(
N
(
N
+
1
)
−
6
K
(
N
−
K
)
−
6
n
(
N
−
n
)
)
+
{\displaystyle {\Big [}(N-1)N^{2}{\Big (}N(N+1)-6K(N-K)-6n(N-n){\Big )}+}
6
n
K
(
N
−
K
)
(
N
−
n
)
(
5
N
−
6
)
]
{\displaystyle 6nK(N-K)(N-n)(5N-6){\Big ]}}
Momentit generoiva funktio
(
N
−
K
n
)
2
F
1
(
−
n
,
−
K
;
N
−
K
−
n
+
1
;
e
t
)
(
N
n
)
{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{t})}}{N \choose n}}\,\!}
Karakteristinen funktio
(
N
−
K
n
)
2
F
1
(
−
n
,
−
K
;
N
−
K
−
n
+
1
;
e
i
t
)
(
N
n
)
{\displaystyle {\frac {{N-K \choose n}\scriptstyle {\,_{2}F_{1}(-n,-K;N-K-n+1;e^{it})}}{N \choose n}}}
Hypergeometrinen jakauma on palauttamattomassa otannassa määrätyn osajoukon esiintymisten jakauma .
Hypergeometrinen jakauma on diskreetti. Jos satunnaismuuttuja
X
{\displaystyle X}
hypergeometrisesti jakautunut , merkitään
X
∼
Hyperg
(
N
,
M
,
n
)
.
{\displaystyle X\sim \operatorname {Hyperg} (N,M,n).}
Parametri
N
{\displaystyle N}
M
{\displaystyle M}
n
{\displaystyle n}
{
0
,
1
,
.
.
.
,
n
}
{\displaystyle \{0,1,...,n\}}
Pistetodennäköisyysfunktio on
P
(
X
=
i
)
=
(
M
i
)
(
N
−
M
n
−
i
)
(
N
n
)
.
{\displaystyle \operatorname {P} (X=i)={\frac {{M \choose i}{N-M \choose n-i}}{N \choose n}}.}
Odotusarvo ja varianssi ovat
E
(
X
)
=
n
M
N
{\displaystyle \operatorname {E} (X)={\frac {nM}{N}}}
Var
(
X
)
=
(
N
−
n
)
n
M
(
N
−
M
)
(
N
−
1
)
N
2
.
{\displaystyle \operatorname {Var} (X)={\frac {(N-n)nM(N-M)}{(N-1)N^{2}}}.}
Diskreettejä jakaumia
Jatkuvia jakaumia
Moniulotteisia jakaumia
![{\displaystyle 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}} \over {N \choose K}}\,_{3}F_{2}\!\!\left[{\begin{array}{c}1,\ k+1-K,\ k+1-n\\k+2,\ N+k+2-K-n\end{array}};1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7787e9d097393f235df082980da2a8a493a44ce)
![{\displaystyle {\frac {(N-2K)(N-1)^{\frac {1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac {1}{2}}(N-2)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ec1b0c28225251fa3fd794e30bffc3eb34315e)
![{\displaystyle 6nK(N-K)(N-n)(5N-6){\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961bbad35da7cd59eb05fb969e0f817f9b1e254c)
