$X$ and $Y$ are jointly distributed continuous random variables,where $Y$ is positive valued and $E[Y^2]=6$. If the conditional distribution of $X$ given $Y=y$ is $U(1-y,1+y)$,then find $Var(X)$.
The only result I’ve been able to conclude from the given information is that $E[X]=1$. I don’t know how to proceed. Do help.
We have, $f_{\small X\mid Y}(x\mid y) =\frac{1}{2y}~\big[ 1-y<x<1+y\big]$
Then $E[X\mid Y]$ comes out to be $1$.
And $E[X^2\mid Y]$ comes out to be $\frac{Y^2}{3} + 1$.
Hence,$V(X\mid Y)=\frac{Y^2}{3} + 1 - 1^2$
Thus,$$\begin{align}V[X] &=E[V(X\mid Y)] + V[E(X\mid Y)] \\&=E[\tfrac{Y^2}{3}] + V(1) \\&=\frac{6}{3} + 0\\&=2\end{align}$$