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Math Help - Probability density function problem

  1. #1
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    Probability density function problem

    Here is a question that I encounter today.

    two continuous random variables A and B whose joint pdf is given by
    f(a,b)=b/(e^(a+b)), a>0, b>0.

    Question
    1.find pdf of U=2-B^2
    2.find the pdf of W=min(A,B)

    Wonder could anybody help?
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  2. #2
    Super Member Anonymous1's Avatar
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    (1) Find the marginal f_B, now do the transform.

    (2) F_W(w) = 1-P(A>w,B>w), and \frac{d}{dx}F_W = f_W.
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  3. #3
    MHF Contributor matheagle's Avatar
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    The rvs A and B are independent by inspection and their marginals are

    f_A(a)=e^{-a}I(a>0) which is an exponential

    and

    f_B(b)=be^{-b}I(b>0) which is a Gamma
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  4. #4
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by Anonymous1 View Post
    (1) Find the marginal f_B, now do the transform.

    (2) F_W(w) = 1-P(A>w,B>w), and \frac{d}{dw}F_W = f_W.

    and

    F_W(w) = 1-P(A>w,B>w)=1-F_A(w)F_B(w)

    and \frac{d}{dw}F_W(w) = f_W(w).
    Last edited by matheagle; May 26th 2010 at 08:39 PM.
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