Question: A point is chosen uniformly at random from the triangle in the plane with vertices (0,0), (1,0), (0,2). Let X be the x-coordinate of this point, and let Y be the y-coordinate of this point.
$$Compute: E(XY^2)$$
I know that $E(XY^2) = E(X)*E(Y^2)$ but I don't know how to compute these two numbers. Could anyone point me in the right direction?
Your $(X,Y)$ are distributed uniform over that triangle $T$. Can you write down their pdf $f(x,y)$? (Remember that $\iint_T f(x,y) dxdy = 1.$)
After you do that, the result is $$ \mathbb{E}\left[XY^2\right] = \iint_T xy^2 f(x,y)dxdy. $$