conditional on X=x Y is binomial(X,p), What is E(exp(y)/X=x)

Question

Suppose we know conditional on X=x Y is binomial(x,p) where p is known.What is E(exp(y)/X=x) where exp is the expodential function and E the expectation
Any help will be appreciated
Thank, You

Answer 1

It looks as if you want to find the expectation of $\exp(Y)$, where $Y$ is binomial, number of trials $x$, probability of success on any trial equal to $p$. To use more familiar notation, we write $n$ instead of $x$. Then $$E(\exp(Y))=\sum_{k=0}^n e^k \binom{n}{k} p^k(1-p)^{n-k}.$$ Rewrite this as $$\sum_{k=0}^n \binom{n}{k} (pe)^k (1-p)^{n-k}.$$ Using the Binomial Theorem, we recognize the above sum as $\left(ep +(1-p)\right)^n$.