# Joint PDF of dependent random variables

How would you solve the below

Let X and Y be random variables with joint probability density function (PDF) given by:

$$f_{X,Y}(x,y) = \begin{cases}cxy &: x \ge 0, y \ge 0, x + y \le 2\\ 0&: \text{otherwise}\end{cases}$$

(a) Find $c$.

(b) Find $\Bbb E [XY ]$.

• Seems like a homework question! tell us what you've figured out so far and where you are stuck. – Ehsan M. Kermani Oct 11 '13 at 22:24
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Draw a picture. Let $T$ be the triangle with corners $(0,0)$, $(2,0)$, $(0,2)$. We want to choose $c$ so that $$\iint_T cxy \,dy\,dx=1.$$ To evaluate the integral, express it as the iterated integral $$\int_{x=0}^2\left(\int_{y=0}^{2-x} cxy\,dy\right)\,dx.$$
For $E(XY)$, we want $$\iint_T (xy)(cxy)\,dy\,dx.$$
• The answer says what to do. Integrate $xyf_{X,Y}(x,y)$ over the plane, which comes down to integrating $(xy)(cxy)$ over the triangle. By the way, I do not get $1$ for $c$. The first integral is $cx(2-x)^2/2)=c(2x-2x^2 +x^3/2)$. Integrating with respect to $x$ from $0$ to $2$ gives $2c/3$, so $c=3/2$. Do check your work, and please note that I could have made an arithmetical error. – André Nicolas Oct 12 '13 at 1:32