Math Help - Conditional Expectation

1. 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!

p.s. sorry for my poor inputting, im a newbie!

2. 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 $E\biggl(\sum_{k=1}^N W_k\biggr)=E(N)E(W_1)$.

3. thanks!

4. 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 $E\biggl(\sum_{k=1}^N W_k\biggr)=E(N)E(W_1)$.
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?