I'm having trouble figuring out the expected value of xy.
Random variables x and y are described by the joint PDF:
$$f(x,y) = \begin{cases} K & \text{: if } x + y ≤ 1 , x > 0 , y > 0 \\ 0 & \text{: otherwise}\end{cases}$$
Determine the expected value of a random variable $r$ defined as $r=x y$
given that $\max(x,y) \leq 0.5$
I've tried to do: $\displaystyle \mathbb{E}(XY) = \iint_{\Omega} xy\cdot f(x,y) \,\mathrm{d}x\,\mathrm{d}y$, but I don't think it's correct.
Best Answer
The integral of f over the region where it is non-zero is K * the area of the right triangle
with sides both of length 1 and so is K * 1/2. This must be 1, so K = 2.
The expected value of X Y given that X and Y are both <= 1/2 is he double integral of x y * 2
over the the square where x, y are <= 1/2. This is 2 * x^2 / 2 * y^2 / 2 for x = y = 1/2.
So the final answer is 1/32.