Consider the following joint PDF for random variables $X$ and $Y$:
(the height that the shading going up to on the $y$-axis is $0.5$, it just didn't show up for some reason).
I'm trying to find $E(X)$ and $E(Y)$ from this joint PDF, but I'm having a little trouble doing so since the shaded ares are not rectangles so it can't be easily calculated. Help me!
Best Answer
We interpret the picture as follows: the joint distribution is uniform on the shaded region. Then the joint density is $4$ on the shaded region, and $0$ elsewhere.
For the expectation of $X$, there is no reason to compute. Symmetry shows that $E(X)=0.5$.
For the expectation of $Y$, we could also use geometry, but let us integrate. So we want to find $$\iint_S 4y\,dx\,dy,$$ where $S$ is the shaded region. We can split the integral into two parts, the left part and the right part. For the left part, $x$ goes from $0$ to $y$, and then $y$ goes from $0$ to $0.5$. We get fairly quickly that the double integral is $\frac{1}{6}$.
We can set up and evaluate a similar but more complicated integral for the right part. But this is unnecessary, since the geometry shows that the integral over the right part is also $\frac{1}{6}$, for a total of $\frac{1}{3}$.
Alternately, we could find the (marginal) distribution of $Y$, and then use that to find $E(Y)$. I prefer the double integral approach.