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Expectation and variance of a random number of random variables Watch

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    I'm really struggling with this question:

    Let X1...XN be independent and identically distributed random variables, N is a non-negative integer valued random variable. Let Z = X1 + ... + XN (assume when N=0 Z=0).

    1. Find E(Z)

    2. Show var(Z) = var(N)E(X1)2 + E(N)var(X1)


    E(Z) = EX (E(X|Z))
    Law of total variance: var(Z) = EX (var(Z|X)) + VarX (E(Z|X))

    1. I think I have managed this, I got E(N)E(X)

    2. I'm unsure how to tackle this one, I know var(Z) = E(Z2) - E(Z)2, and I know E(Z)2 but I don't know how to calculate the other, or if I should be using the equation above, and if so, how.
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    When i =/= j, X_i and X_j are independent, and so E[X_i X_j] = E[X_i]E[X_j] = E[X]^2
    When i = j, E[X_i X_j] = E[X^2] = Var[X]+E[X]^2

    Use this to find a simple expression for E[(X_1+X_2+...+X_n)^2] (for fixed n) and then use this to find E[Z^2].
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    Let X1,...,XN be independent identically distributed random variables, where N is a
    non-negative integer valued random variable. Let Z = X1 + ... + XN, assuming that Z = 0 if N = 0.

    Find E(Z) and show that


    var(Z) = var(N)E(X1)2 + E(N)var(X1)
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    (Original post by The Muon)
    Let X1,...,XN be independent identically distributed random variables...
    See above...
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    (Original post by DFranklin)
    See above...
    My bad for the posting the same topic, should learn to use the search :facepalm2:
 
 
 
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