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**mistykz** Suppose x,y are uniformly distributed over the region 0=< X =< Y =< 1

Prove that E(x|Y=y) = y/2 and E(Y|X=x) = x + (1-x)/2

So, I found the density function f(x,y) to be equal to 2, as long as x and y are within the bounds.

I set up E(x|Y=y) = integral from 0 to y of x*f(x|y) dx

f(x|y) = f(x,y)/f(y)

f(y) = integral from 0 to 1 of 2 dx, therefore 2.

Then f(x|y) = 2/2 = 1

And E(x|Y=y) = integral from 0 to y of x * dx, but then that would give (y^2)/2, which doesn't appear to be correct...can anyone help me please? I'm slightly befuddled. Also, would the second one go the same way?

Thank you!