Actually my question arises from the definition of $E[XY]$, why is it defined as the integral of $xyf(x,y)$?

If so, what is the expectation of $xy^2$??

Thank you very much.

  • $\begingroup$ It is not precisely defined in that way. The random variable $H(X,Y)$ is, well, a random variable $W$ and the expectation of $W$ is defined in the usual way, using the distribution function of $W$. However, it is a theorem that under reasonable conditions on $H$, the expectation of $W$ is the integral you gave. In elementary courses, it is often asserted without proper proof that the expectation of $H(X,Y)$ is what you wrote down. $\endgroup$ Jul 9, 2013 at 15:06

1 Answer 1


The expected value of $g(X,Y)$ in general is

$$\iint_R dx dy \, g(x,y) f(x,y)$$

where $R$ is the region over which the random variables $X$ and $Y$ are defined to take on values. Therefore

$$E(X Y^2) = \iint_R dx dy \, x y^2 f(x,y)$$

This of course assumes that $f$ is properly normalized as a pdf.

  • 2
    $\begingroup$ @WayneN: please do not change the meaning of my posts through an edit. Your previous edit demonstrated that you either did not read or did not understand what I wrote. $\endgroup$
    – Ron Gordon
    Jul 9, 2013 at 15:05
  • $\begingroup$ Could I ask why the expectation is defined in this way? The formula of E[X+Y]=E[X]+E[Y] only holds true when x and y are independent. Is that correct? $\endgroup$ Jul 10, 2013 at 9:39

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .