Solution Found!
Suppose that Y has a binomial distribution with parameters
Chapter 5, Problem 142E(choose chapter or problem)
Suppose that \(Y\) has a binomial distribution with parameters \(n \text { and } p\) but that \(p\) varies from day to day according to a beta distribution with parameters \(\alpha \text { and } \beta\). Show that
a \(E(Y)=n \alpha /(\alpha+\beta)\)
b \(V(Y)=\frac{n \alpha \beta(\alpha+\beta+n)}{(\alpha+\beta)^{2}(\alpha+\beta+1)}\)
Equation Transcription:
Text Transcription:
Y
n and p
p
\alpha and \beta
E(Y)=n \alpha /(\alpha+\beta)
V(Y)=\frac n \alpha \beta(\alpha+\beta+n)(\alpha+\beta)^2(\alpha+\beta+1)
Questions & Answers
QUESTION:
Suppose that \(Y\) has a binomial distribution with parameters \(n \text { and } p\) but that \(p\) varies from day to day according to a beta distribution with parameters \(\alpha \text { and } \beta\). Show that
a \(E(Y)=n \alpha /(\alpha+\beta)\)
b \(V(Y)=\frac{n \alpha \beta(\alpha+\beta+n)}{(\alpha+\beta)^{2}(\alpha+\beta+1)}\)
Equation Transcription:
Text Transcription:
Y
n and p
p
\alpha and \beta
E(Y)=n \alpha /(\alpha+\beta)
V(Y)=\frac n \alpha \beta(\alpha+\beta+n)(\alpha+\beta)^2(\alpha+\beta+1)
ANSWER:Solution :
Step 1 of 1:
Let Y follows Binomial distribution with parameters n and p.
Our goal is:
a). We need to show that E(Y) = .
b). We need to show that V(Y) = .
a).
Now we have to show that E(Y) = .
The formula for binomial distribution is
P(X=k) =
Beta distribution is
f(y) =
The conditional expectation is
E(gg
Then we obtain
E() =
Then E(y/p) is