# Math Help - Expectation

1. ## 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!)

2. 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

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