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Thread: Calculating and/or estimating moments: Let X_1,...X_n be IID pos. valued RV...

  1. #1
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    Feb 2018
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    Calculating and/or estimating moments: Let X_1,...X_n be IID pos. valued RV...

    I dont know how to compute or prove the following

    Let X_1,...X_n be IID pos. valued RV and P(X_i > x) = 1\(1+x^a) for all i=1,....n, and let Y_n = max{X1,....,X_n}

    First, for any natural number m, whats the condition of "a" so that E[Y_2^m] < inf, so basically when are the moments of Y_2 finit.
    P(Y_2<x) should be (x^(2a))/(1+x^a)^2, but I dont know how to integrate that...? (Or is there another efficent method to show it?)

    Second, how to show the equality between E(Y_n^m) < inf <=> E(X_1^m) < inf.

    Any help much appreciated!

  2. #2
    New Member
    Join Date
    Feb 2018
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    2

    Finitness of moments

    Hey guys, Im having some trouble with the following.

    Let (X_1,...,X_n) be IID pos. valued RV and P(X_i > x) = 1/(1+x^a) for all i=1,....n, x>0
    Put Y_n = max{X_1,...X_n}.

    First for what "a" is E(Y_n^2) < inf for any natural m ? I tried to calculate the prob. density of Y by differentiating P(Y_2<x) = (x^2a)/(1+x^a)^2, but the terms get huge and unsolvable for me...

    Another problem I struggle to prove is the equality E(Y_n^m) < inf <=> E(X_1^m) < inf.

    I think I can show <= by the monotony of the integral, but I dont know how to proceed for =>.

    Any help is much appreciated!

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