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    • Thread Starter

    I need some help on this question:

    We have that X1 and X2 are independent random variables. They each have the same probability density, f(x) = 4(x)^3 for 0<x<1 and 0 otherwise.

    We are also given that Y1 and Y2 are random variables defined as Y1=X1(X2)^0.5 and Y2=X2(X1)^0.5.

    The question is to find the joint density of Y1 and Y2, and their joint densities.

    I found the joint density to be (64(Y1)^5/3 (Y2)^5/3)/3. However, my problem is that I'm not sure how to find the regions where the density is applicable (i.e like 0<x<1) for the initial density function. I
    I've tried subbing in expressions for X1 and X2 in terms of Y1 and Y2 into 0<X1<1 and 0<X2<1 but this seems to arrive at a contradiction as I get 0<Y1<(Y2)^0.5 and 0<Y2<(Y1)^0.5. Could anyone help me out please?
    • Study Helper

    Study Helper
    (Original post by pineapplechemist)
    Could anyone help me out please?
    An interesting question.

    Note: What you have is not contradictory, since 0<Y1,Y2<1

    It's clear 0 < Y1,Y2 < 1, but is the domain equal to (0,1) X (0,1) ?

    I got Y_1 &gt; Y_2^{5/2}&gt;0 and similarly Y2,Y1.

    Yours is the stronger restriction, so I may have missed a trick, or slipped up.
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Updated: October 30, 2015
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