Calculate $E[Y]$ and $Var(Y|X)$

Question

Suppose $X∼ U(0,1)$ and that conditional on $X$ being observed, $Y|X =x ∼ U(0,x)$. Calculate $E[Y]$.

To find $E[Y]$, I would like to use the proposition in my textbook that says $E[Y]=E[E[Y|X]]$.

However, I get that $E[Y|X] = \int_{0}^xxf(y|x)dx=\int_{0}^xx(1/x)dx=x$, while my professor told us that $E[Y|X]$ should equal $x/2$.

Could someone tell me what I did wrong? Thank you

Also, how would you find $var(Y|X)$?

Answer 1

The PDF for a $U(0,x)$ random variable is $1/x.$

Hence,

$$E[Y|X=x] = \int_0^x t (1/x) \, dt = x/2.$$