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Math Help - Mean and variance of two stochastic variables combined

  1. #1
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    Cool Mean and variance of two stochastic variables combined

    Hi,

    I'm stuck with a little problem that I need to use for my research.

    Let X and N be stochastic random variables. I need to show that:

    Mean:
    E(sum(X)) = E(N) * E(X)

    Variance:
    var(sum(X)) = E(N) * var(X) + var(N) * (E(X))^2

    Anyone knowing to do this?

    Cheers, Eva
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  2. #2
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by Eva BSc View Post
    Mean:
    E(sum(X)) = E(N) * E(X)
    When N=n, \ \ S = \sum_i X_i has E(S)=nE(X_i)

    Breaking things down according to the value of N, we have...

    E[S] = \sum_{n=0}^{\infty} E(S|N=n)P(N=n) = \sum_{n=0}^{\infty} nE(X_i)P(N=n) = E(N)E(X_i)
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  3. #3
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    Quote Originally Posted by Eva BSc View Post
    Variance:
    var(sum(X)) = E(N) * var(X) + var(N) * (E(X))^2
    When N=n, \ \ S = \sum_i X_i has Var(S) = nVar(X_i) and hence,

    E(S^2|N=n)= nVar(X_i) + (nE(X_i))^2

    Then,

    E[S^2] = \sum_{n=0}^{\infty} E(S^2|N=n)P(N=n) = \sum_{n=0}^{\infty} \{nVar(X_i) + n^2 (E(X_i))^2 \}P(N=n) = E(N)Var(X_i) + E(N^2) (E(X_i))^2

    So,

    Var(S) = E(S^2) - (E(S))^2 = E(N)Var(X_i) + E(N^2)(E(X_i))^2 - (E(N)E(X_i))^2 = E(N)Var(X_i) + Var(N)(E(X_i))^2
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    Thnx!
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    Super Member Anonymous1's Avatar
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    I had to prove this in my homework a while back.
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  6. #6
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    I've used this fact before, but never seen its proof. Thanks!
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  7. #7
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  8. #8
    Super Member Anonymous1's Avatar
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    Just to confirm, my instructor verified that the proof above is valid and correct...
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