Conditional Expectation Question (Probability Theory)

[Sprng]

Homework Statement

(Question is #6 on p.171 in An Introduction to Probability and Statistics by Ruhatgi & Saleh)

Let X have PMF P_λ{X=x} = λ^xe^-λ/x!, x=0,1,2...
and suppose that λ is a realization of a RV Λ with PDF
f(λ)=e^-λ, λ>0.

Find E(e^-Λ|X=1)

The Attempt at a Solution

The answer according to the solutions in the back of the text is 4/9.

I'm really not sure where to begin. My problem is that one of these distributions is continuous and one is discrete and there are no conditional probability examples that are like that.

What I know is that X~Poisson and Λ~Exponential.

I believe that E(e^-Λ|X=1)=integral{e^-Λf_Λ|X(λ|x)dλ,from λ=0 to infinity}. However, I do not see an easy way to calculate f_Λ|X(λ|x).

Also, would the first distribution be considered the conditional probability mass function of X given λ? (That's how I was treating it in my attempts).

Thanks in advance for any tips.

 

[Sprng]

Yay. Nevermind. I was somehow able to solve it. I found the marginal distribution of X, and replaced the fλ|X with fX|λ(x|λ)*f(λ)/f(x) and it worked. :)