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Math Help - Continuous random variable

  1. #1
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    Continuous random variable

    I'm stuck at this question, if anyone could help please.

     f XY(x,y) = kxy,  0 < x < 1, 0 <y < 1, 0 < x + y < 1

    Show k = 24, -- I proved this by taking the double integral and setting = 1.
    Find the marginal probility density function of X and Y,
    Find the pdf of U = X + Y, X + Y <= u ( 0 < u < 1)

    My marginal pdf for X was  12x(1-x)^2 , 0 < x < 1
    and for Y, was  12y(1-y)^2 , 0 < y < 1

    Is this correct, and if so how would I find U = X + Y, do I integrate (x+y)*24xy

    Also my epected value for E(X) and E(Y) was = 1/5
    Last edited by mathsandphysics; February 18th 2011 at 01:29 AM.
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  2. #2
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    Quote Originally Posted by mathsandphysics View Post
    I'm stuck at this question, if anyone could help please.

     f XY(x,y) = kxy, 0 < x < 1, 0 <y < 1, 0 < x + y < 1

    Show k = 24, -- I proved this by taking the double integral and setting = 1.
    Find the marginal probility density function of X and Y,
    Find the pdf of U = X + Y, X + Y <= u ( 0 < u < 1)

    My marginal pdf for X was  12x(1-x)^2 , 0 < x < 1
    and for Y, was  12y(1-y)^2 , 0 < y < 1

    Is this correct, and if so how would I find U = X + Y, do I integrate (x+y)*24xy

    Also my epected value for E(X) and E(Y) was = 1/5
    Your marginal distributions are correct. But I get E(X) = E(Y) = 2/5.

    There are several options for finding the pdf of U = X + Y.

    One option is to use the Change of Variable Theorem (see for example Walpole, Myers and Myers - Probability and Statistics For Engineers and Scientists): Let U = X + Y and V = X. Get the joint pdf of U and V and then calculate the marginal pdf for U. I get g(u) = 4u^3.

    Another option (more work) would be to use the method of distribution functions (see for example Mendenhall, Scheaffer and Wackerley - Mathematical Statistics With Applications).
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  3. #3
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    Thanks, I made a typo for E(X), I got 2 /5 too.

    I dont think we have covered these so far, so I'm gonna try the chage of variables.
    Thanks
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