I am practising some exam questions and am failing to understand the problem at hand. I believe I am supposed to take the double integral of the joint PDF that can be calculated by noting that $f_X,_Y(x,y) = f_{X|Y}(x)*f_Y(y)$.
The problem is as such:
The conditional pdf for X given Y = y, is
$
f_{X|Y=y}(x) =
\begin{cases}
1/y^2, & \text{for 0 $\leq$ x $\leq$ $y^2$} \\
0, & \text{otherwise,}
\end{cases}
$
While the marginal density of Y is
$
f_Y(y) =
\begin{cases}
4y^3, & \text{for 0 $\leq$ y $\leq$ 1} \\
0, & \text{otherwise.}
\end{cases}
$
Now I think that X and Y are not independent, this is because looking at the limits of $f_{X|Y}(x)$ it is clear that if y = 0 then x must be 0. Hence, I need to double integrate over the joint pdf to find E(XY), I assume. The problem is how do I determine the limits of my integral?
Thanks for your patience, help and time! It is much appreciated!
Best Answer
The limits of the integral are in fact given to you. They are: $0\leq y\leq 1$ and $0\leq x\leq y^2$.
$$\begin{align} \mathsf E(XY) & = \int_{y=0}^1 \int_{x=0}^{y^2} x\,y\; f_Y(y)\;f_{X\mid Y=y}(x)\operatorname d x\operatorname d y \\[1ex] & = \int_{y=0}^1 y\cdot 4y^3/y^2 \int_{x=0}^{y^2} x\operatorname d x\operatorname d y \end{align}$$