# Conditional Expectation

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• Mar 13th 2009, 09:07 AM
Roro860
Conditional Expectation
I need to solve this problem using conditional expectation:

A hen lays N eggs where N is Poisson with parameter k. The weight of the nth egg is Wn where W1, W2, ... are independent identically distributed random variables with mean x. Let W=sum from i=1 to i=N of Wi.

Show E[W]=xk

Any help would be hugely grateful! (Smile)

p.s. sorry for my poor inputting, im a newbie!
• Mar 13th 2009, 02:51 PM
matheagle
You can use Wald's Equation directly or copy the proof for your case...
Wald's equation - Wikipedia, the free encyclopedia
(Hint: Use proof 2)
But Wald says that $\displaystyle E\biggl(\sum_{k=1}^N W_k\biggr)=E(N)E(W_1)$.
• Mar 13th 2009, 10:59 PM
Roro860
thanks! :)
• May 29th 2009, 02:11 PM
ilgatto69
Quote:

Originally Posted by matheagle
You can use Wald's Equation directly or copy the proof for your case...
Wald's equation - Wikipedia, the free encyclopedia
(Hint: Use proof 2)
But Wald says that $\displaystyle E\biggl(\sum_{k=1}^N W_k\biggr)=E(N)E(W_1)$.

It seems that if N is also independent, you don't need Wald's equation, just the simple rules of conditional expectation are enough.... (see Compound Poisson process - Wikipedia, the free encyclopedia and Law of total expectation - Wikipedia, the free encyclopedia)

I'm trying to figure out where the independence of N is needed in this elementary proof, but so far I haven't been able. But it must be needed, otherwise Wald's equation wouldn't have the quite complex proof it has.

Someone can help me?