Suppose, Χ | θ ~ Binom(m,0)and 0 ~ Beta(α,β), then E(X) is equals to,
Answers
Answer:
n ∈ N0 — number of trials
{\displaystyle \alpha >0}\alpha >0 (real)
{\displaystyle \beta >0}\beta >0 (real)
Support
k ∈ { 0, …, n }
PMF
{\displaystyle {\binom {n}{k}}{\frac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}\!}{\displaystyle {\binom {n}{k}}{\frac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}\!}
CDF
{\displaystyle {\begin{cases}0,&k<0\\{\binom {n}{k}}{\tfrac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}{}_{3}\!F_{2}({\boldsymbol {a}},{\boldsymbol {b}},k),&0\leq k<n\\1,&k\geq n\end{cases}}}{\displaystyle {\begin{cases}0,&k<0\\{\binom {n}{k}}{\tfrac {\mathrm {B} (k+\alpha ,n-k+\beta )}{\mathrm {B} (\alpha ,\beta )}}{}_{3}\!F_{2}({\boldsymbol {a}},{\boldsymbol {b}},k),&0\leq k<n\\1,&k\geq n\end{cases}}}
where 3F2(a,b,k) is the generalized hypergeometric function
{\displaystyle {}_{3}\!F_{2}(1,-k,n\!-\!k\!+\!\beta ;n\!-\!k\!-\!1,1\!-\!k\!-\!\alpha ;1)\!}{\displaystyle {}_{3}\!F_{2}(1,-k,n\!-\!k\!+\!\beta ;n\!-\!k\!-\!1,1\!-\!k\!-\!\alpha ;1)\!}
Mean
{\displaystyle {\frac {n\alpha }{\alpha +\beta }}\!}{\frac {n\alpha }{\alpha +\beta }}\!
Variance
{\displaystyle {\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!}{\frac {n\alpha \beta (\alpha +\beta +n)}{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!
Skewness
{\displaystyle {\tfrac {(\alpha +\beta +2n)(\beta -\alpha )}{(\alpha +\beta +2)}}{\sqrt {\tfrac {1+\alpha +\beta }{n\alpha \beta (n+\alpha +\beta )}}}\!}{\tfrac {(\alpha +\beta +2n)(\beta -\alpha )}{(\alpha +\beta +2)}}{\sqrt {{\tfrac {1+\alpha +\beta }{n\alpha \beta (n+\alpha +\beta )}}}}\!
Ex. kurtosis
See text
MGF
{\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{t})\!}_{{2}}F_{{1}}(-n,\alpha ;\alpha +\beta ;1-e^{{t}})\! where {\displaystyle _{2}F_{1}}{\displaystyle _{2}F_{1}} is the hypergeometric function
CF
{\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-e^{it})\!}_{{2}}F_{{1}}(-n,\alpha ;\alpha +\beta ;1-e^{{it}})\!
PGF
{\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-z)\!}{\displaystyle _{2}F_{1}(-n,\alpha ;\alpha +\beta ;1-z)\!}