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Math Help - Order statistics - determine the cdf

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    Order statistics - determine the cdf

    Let u_1,...,u_N be N independent random variables uniformly distributed on the interval  [0,1] . Determine the cdf of y:= max_n u_n and z := min_n u_n

    So we are after the cdf of F_y(s)=P(max_u \leq s) which is the probability that max_u is greater then s_1, s_2...? not sure how to formalise/what distribution this is?

    Thanks for any help..
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    Quote Originally Posted by Robb View Post
    Let u_1,...,u_N be N independent random variables uniformly distributed on the interval  [0,1] . Determine the cdf of y:= max_n u_n and z := min_n u_n

    So we are after the cdf of F_y(s)=P(max_u \leq s) which is the probability that max_u is greater then s_1, s_2...? not sure how to formalise/what distribution this is?

    Thanks for any help..
    Hint:

    max \{u_1, u_2, \dots , u_n\} \leq y
    if and only if
    u_i \leq y for all i = 1, 2, \dots , n.
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    Quote Originally Posted by Robb View Post
    Let u_1,...,u_N be N independent random variables uniformly distributed on the interval  [0,1] . Determine the cdf of y:= max_n u_n and z := min_n u_n

    So we are after the cdf of F_y(s)=P(max_u \leq s) which is the probability that max_u is greater then s_1, s_2...? not sure how to formalise/what distribution this is?

    Thanks for any help..
    You will find detailed proofs in most mathematical statistics books, using Google and also by searching this subforum (use the Search tool).
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    hmm, so after doing a bit of research...
    The distribution of independent order statistics would be;
    For the Maximum
    F_{Y_{N}}(s)=P(Y_{n}\leq s)=P(Y_1\leq s)P(Y_2\leq s)...P(Y_n\leq s)=[F_U(s)]^N
    So since since it is the uniform distribution with F_U(s)=s the cdf is F_{Y_{N}}(s)=s^N

    And the minimum will be
    F_{Z_1}(s)=1-P(Z_1 > s)=1-[1-F_U(s)]^N
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    Quote Originally Posted by Robb View Post
    hmm, so after doing a bit of research...
    The distribution of independent order statistics would be;
    For the Maximum
    F_{Y_{N}}(s)=P(Y_{n}\leq s)=P(Y_1\leq s)P(Y_2\leq s)...P(Y_n\leq s)=[F_U(s)]^N
    So since since it is the uniform distribution with F_U(s)=s the cdf is F_{Y_{N}}(s)=s^N

    And the minimum will be
    F_{Z_1}(s)=1-P(Z_1 > s)=1-[1-F_U(s)]^N
    Order Statistics 10/30
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