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Math Help - Distribution functions

  1. #1
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    Distribution functions

    Hi there,

    I was wondering if someone could explain how to go about the following problem.

    Let X and Y be independent random variables having distribution functions Fx and FY , respectively.

    a) Define Z = max{X,Y} to be the larger of the two. Show that FZ(z) = FX(z)FY(z) for all z.

    b) Define W = min{X,Y} to be the smaller of the two. Show that FW(w) = 1 - [1-FX(w)][1-FY(w)] for all w.

    Thanks!
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  2. #2
    MHF Contributor harish21's Avatar
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    Re: Distribution functions

    Quote Originally Posted by supermario88 View Post
    Hi there,

    I was wondering if someone could explain how to go about the following problem.

    Let X and Y be independent random variables having distribution functions Fx and FY , respectively.

    a) Define Z = max{X,Y} to be the larger of the two. Show that FZ(z) = FX(z)FY(z) for all z.

    b) Define W = min{X,Y} to be the smaller of the two. Show that FW(w) = 1 - [1-FX(w)][1-FY(w)] for all w.

    Thanks!
    since they are independent,

    F_Z(z)=P[Z \leq z]=P[(X \leq z)\cap (Y \leq z)]= P(X \leq z) P(Y \leq z)=F_X(z)\cdot F_Y(z)

    and

    F_W(w)=P[W \leq w]=1-P[(X>w) \cap (Y>w)]=.......
    Last edited by harish21; September 2nd 2012 at 08:45 AM.
    Thanks from supermario88
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  3. #3
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    Re: Distribution functions

    Hi harish21,

    Thanks very much for the reply. So I can finish letter b thanks to your hint. Would you mind explaining the intuition of letter a? Specifically, could you talk about why

    P[Z≤z] = P[(X≤z)∩(Y≤z)]

    I just want to develop a better understanding of the problem. Thank you.
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  4. #4
    MHF Contributor matheagle's Avatar
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    Re: Distribution functions

    Z is the largest of X and Y
    so IF Z is less than or equal to a, then both X and Y must be less than or equal to a.
    For the minimum, use the complement twice
    Thanks from harish21 and supermario88
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  5. #5
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    Re: Distribution functions

    I see. Thanks very much!
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  6. #6
    MHF Contributor matheagle's Avatar
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    Re: Distribution functions

    I work with order statistics all the time.
    I can google my name and find some of my papers...
    http://w3.math.sinica.edu.tw/bulleti...d322/32203.pdf
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  7. #7
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    Re: Distribution functions

    Understood. Thanks matheagle.
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