از ویکیپدیا، دانشنامهٔ آزاد
توزیع دریکله عمومی یک توزیع پیوسته در آمار است که عمومی شده توزیع دریکله است و به اندازه دو برابر آن پارامتر دارد.
تابع چگالی
p
1
,
…
,
p
k
−
1
{\displaystyle p_{1},\ldots ,p_{k-1}}
[
∏
i
=
1
k
−
1
B
(
a
i
,
b
i
)
]
−
1
p
k
b
k
−
1
−
1
∏
i
=
1
k
−
1
[
p
i
a
i
−
1
(
∑
j
=
i
k
p
j
)
b
i
−
1
−
(
a
i
+
b
i
)
]
{\displaystyle \left[\prod _{i=1}^{k-1}B(a_{i},b_{i})\right]^{-1}p_{k}^{b_{k-1}-1}\prod _{i=1}^{k-1}\left[p_{i}^{a_{i}-1}\left(\sum _{j=i}^{k}p_{j}\right)^{b_{i-1}-(a_{i}+b_{i})}\right]}
که در آن تعریف میکنیم
p
k
=
1
−
∑
i
=
1
k
−
1
p
i
{\displaystyle p_{k}=1-\sum _{i=1}^{k-1}p_{i}}
تابع عمومی لحظه [ ویرایش ] اگر
X
=
(
X
1
,
…
,
X
k
)
∼
G
D
k
(
α
1
,
…
,
α
k
;
β
1
,
…
,
β
k
)
{\displaystyle X=\left(X_{1},\ldots ,X_{k}\right)\sim GD_{k}\left(\alpha _{1},\ldots ,\alpha _{k};\beta _{1},\ldots ,\beta _{k}\right)}
E
[
X
1
r
1
X
2
r
2
⋯
X
k
r
k
]
=
∏
j
=
1
k
Γ
(
α
j
+
β
j
)
Γ
(
α
j
+
r
j
)
Γ
(
β
j
+
δ
j
)
Γ
(
α
j
)
Γ
(
β
j
)
Γ
(
α
j
+
β
j
+
r
j
+
δ
j
)
{\displaystyle E\left[X_{1}^{r_{1}}X_{2}^{r_{2}}\cdots X_{k}^{r_{k}}\right]=\prod _{j=1}^{k}{\frac {\Gamma \left(\alpha _{j}+\beta _{j}\right)\Gamma \left(\alpha _{j}+r_{j}\right)\Gamma \left(\beta _{j}+\delta _{j}\right)}{\Gamma \left(\alpha _{j}\right)\Gamma \left(\beta _{j}\right)\Gamma \left(\alpha _{j}+\beta _{j}+r_{j}+\delta _{j}\right)}}}
که
δ
j
=
r
j
+
1
+
r
j
+
2
+
⋯
+
r
k
{\displaystyle \delta _{j}=r_{j+1}+r_{j+2}+\cdots +r_{k}}
E
(
X
j
)
=
α
j
α
j
+
β
j
∏
m
=
1
j
−
1
β
m
α
m
+
β
m
.
{\displaystyle E\left(X_{j}\right)={\frac {\alpha _{j}}{\alpha _{j}+\beta _{j}}}\prod _{m=1}^{j-1}{\frac {\beta _{m}}{\alpha _{m}+\beta _{m}}}.}
https://web.archive.org/web/20081119000022/http://www.nscb.gov.ph/ncs/9thncs/papers/theory_Generalization.pdf
![{\displaystyle \left[\prod _{i=1}^{k-1}B(a_{i},b_{i})\right]^{-1}p_{k}^{b_{k-1}-1}\prod _{i=1}^{k-1}\left[p_{i}^{a_{i}-1}\left(\sum _{j=i}^{k}p_{j}\right)^{b_{i-1}-(a_{i}+b_{i})}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c869562ff9a735ae165555eb24ae7cb86252130a)
![{\displaystyle E\left[X_{1}^{r_{1}}X_{2}^{r_{2}}\cdots X_{k}^{r_{k}}\right]=\prod _{j=1}^{k}{\frac {\Gamma \left(\alpha _{j}+\beta _{j}\right)\Gamma \left(\alpha _{j}+r_{j}\right)\Gamma \left(\beta _{j}+\delta _{j}\right)}{\Gamma \left(\alpha _{j}\right)\Gamma \left(\beta _{j}\right)\Gamma \left(\alpha _{j}+\beta _{j}+r_{j}+\delta _{j}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05ce9a8642b88d6bcd8c900778426f4d259b80b5)
