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Math Help - Calculate PDF of Y-Z given Y>Z

  1. #1
    Newbie
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    Mar 2010
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    Calculate PDF of Y-Z given Y>Z

    Hi everyone,

    I am trying to calculate the PDF of Y-Z given that Y>Z. Random variables Y and Z are independent and uniformly distributed over the interval [0,1].

    I have tried calculating |Y-Z| (the PDF is basically a half triangle with height 2 and width 1), but I do not think this is correct. I think that the restriction Y>Z creates a situation where the random variables are not independent, so convolution is not valid.

    In trying to understand this problem, I created a Matlab script to look at the empirical PDF. The result is clearly not linear like we would expect from |Y-Z|, but it has a quadratic form of some kind. The Matlab code is listed below.

    Any ideas for solving this?

    ==========================================

    clear;
    clc;

    out = [];

    for rounds = 1:30000;

    y = rand;
    z = rand;

    while y<z
    z = rand;
    end

    out = [out y-z];

    end

    [pdf,xout] = hist(out,100);
    pdf = pdf/sum(pdf);
    stem(xout,pdf)
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  2. #2
    Super Member
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    Jul 2009
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    Maybe, if you posted the entire problem.
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  3. #3
    Newbie
    Joined
    Mar 2010
    Posts
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    Thanks for your reply. Although it seems simple, this is the entire problem statement.

    Random variables Y and Z are independent and uniformly distributed over the interval [0,1]. Define W=Y-Z and find the PDF f_W(w|Y>Z).
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