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Math Help - Expectation

  1. #1
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    Expectation

    I need to find E[Y/X=x] and E[X/Y=y] for
    f(x,y)=24xy (x,y) is region bounded by y=x^2 and y=x.

    I'm having trouble deciding on the limits of integration for the two relevant integrals:
    yf(y/x)dx and xf(x/y)dy

    how do I find the limits for the integrations above?

    (I apologise for my lack of LaTex!)
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  2. #2
    MHF Contributor matheagle's Avatar
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    I first checked to see if it's a valid density and it is.
    The line y=x is above y=x^2 and the points of intersection are (0,0) to (1,1).

    f(x)=24\int_{x^2}^x xydy=12x(x^2-x^4)

    and

    f(y)=24\int_y^{\sqrt y} xydx=12y(y-y^2)

    Well once you obtain f(x|y) and f(y|x) by division the bounds are easy, it's 0 to 1 in those two integrals.
    That leads me to believe your problems are is obtaining the correct conditional densities.

    Obtain f(x|y) and f(y|x), then

    E(X|Y=y)=\int_0^1 x f(x|y)dx

    and

    E(Y|X=x)=\int_0^1 y f(y|x)dy
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