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Math Help - Order Statistics of Beta Function

  1. #1
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    Order Statistics of Beta Function

    Hey, I need help from you on another problem.

    Let X1, ... , Xn ~(iid, independent and identical distribution) Beta(Alpha, 1)
    (a) Prove that X(n), the maximum of random variable Xi, also has a Beta Distribution and identify its parameter.

    (b) Based on (a), What is E(X(n))?

    I'm very sorry that I don't know how to write equation as others.
    Anyway, let me show your work.

    Let X(n)= A.
    Then, FA(a), F of A of a = [FX(a)]^n , F of X of a to the n.
    And fA(a)=n(FX(a))^(n-1)*fX(a)
    But I can't find FX(a).

    THank you.
    892king
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  2. #2
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    Quote Originally Posted by 892king View Post
    Hey, I need help from you on another problem.

    Let X1, ... , Xn ~(iid, independent and identical distribution) Beta(Alpha, 1)
    (a) Prove that X(n), the maximum of random variable Xi, also has a Beta Distribution and identify its parameter.

    (b) Based on (a), What is E(X(n))?

    I'm very sorry that I don't know how to write equation as others.
    Anyway, let me show your work.

    Let X(n)= A.
    Then, FA(a), F of A of a = [FX(a)]^n , F of X of a to the n.
    And fA(a)=n(FX(a))^(n-1)*fX(a)
    But I can't find FX(a).

    THank you.
    892king
    General result:

    The pdf of X(n) = \text{max} \, (X_1, \, X_2, \, ....... \, X_n) where X_1, \, X_2, \, ....... \, X_n are i.i.d. is f_n(x) = n [F(x)]^{n-1} f(x) where F(x) is the cdf of X and f(x) is the pdf of X.


    For X ~ Beta (\alpha, 1), ~ f(x) = \alpha x^{\alpha - 1}, ~ 0 \leq x \leq 1 and zero elsewhere.

    Therefore F(x) = \alpha \int_0^x t^{\alpha - 1} \, dt = x^{\alpha}, ~ 0 \leq x \leq 1.

    Therefore f_n(x) = n (x^{\alpha})^{n-1} \alpha x^{\alpha - 1} = n \alpha x^{n \alpha - 1}, ~ 0 \leq x \leq 1 and zero elsewhere.

    That is, f_n(x) = \text{Beta} \, (n \alpha, 1).
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