Question

Let the joint pdf of X , Y be , f (x, y) = 2 , 0 ≤ x ≤ y ≤ 1 and

f1(x) = 2(1 − x) , 0 ≤ x ≤ 1. Find E[X|Y]

Answer 1

$E[X|Y]=\int xf(x|y)dx$

$f(x|y)=\frac{f(x,y)}{f(y)}=f(x)$

$E[X|Y]=\int xf(x)dx$

$E[X|Y]=2\int^1_0x(1-x)dx=(x^2-2x^3/3)|^1_0=1-2/3=1/3$