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Math Help - simple questions need answer # 3

  1. #1
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    Question simple questions need answer # 3

    3. Let the joint density of (X,Y) be given by f(x,y)=2, 0<X<1, 0<Y<1-X
    a) Calculate marginal density f(x),
    b) Calculate P(0<Y<3/4|X=0.5).

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    Last edited by mr fantastic; September 8th 2009 at 04:55 AM. Reason: Moved to new thread
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  2. #2
    MHF Contributor matheagle's Avatar
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    the constraint is x+y\le 1

    the two rvs are uniformly distributed over that triangle in the first quadrant

    so f_X(x)=\int_0^{1-x}2dy=2(1-x) on 0<x<1

    f(y|x)= {2\over 2(1-x)}={1\over 1-x} on 0<x+y<1

    THIS IS UNIFORMLY DISTRIBUTED on any line for a fixed x

    f(y|x=.5)= 2 on 0<y<.5

    so P(0<Y<3/4|X=0.5) is a silly question, the answer is 1.

    P(0<Y<3/4|X=0.5) =\int_0^{.75}f(y|x=.5)dy = \int_0^{.5}2dy=1
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