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.
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.