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Math Help - Conditional Expectation

  1. #1
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    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!
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  2. #2
    MHF Contributor matheagle's Avatar
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    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).
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  3. #3
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    thanks!
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  4. #4
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    Quote Originally Posted by matheagle View Post
    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?
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