Conditional Expectation Question (Probability Theory)

  • Thread starter Sprng
  • Start date
  • #1
Sprng
2
0

Homework Statement



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

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

Find E(e|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|X=1)=integral{efΛ|X(λ|x)dλ,from λ=0 to infinity}. However, I do not see an easy way to calculate fΛ|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.
 

Answers and Replies

  • #2
Sprng
2
0
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. :)
 

Suggested for: Conditional Expectation Question (Probability Theory)

Replies
2
Views
413
Replies
4
Views
270
  • Last Post
Replies
16
Views
800
Replies
4
Views
763
Replies
16
Views
423
Replies
4
Views
153
Replies
5
Views
389
Replies
5
Views
441
Replies
6
Views
198
Top