E(XY) expectation product Could someone give an example on when and why you would want to multiply two random variables? What is the reason? For example, what does the expectation E(XY) mean in intuitive terms?
Thanks in advance!
 A: Just as the expectation of one random variable is intuitively the "average" value that you are going to see if you got to see many realizations from its distribution, the expectation of the product of random variables is going to be the "average" value that you are going to see if you get to see many realizations from the distribution Z = XY where X and Y are themselves random variables. So you can think about XY as a random variable in and of itself and then $E(XY) = E(Z)$ as the expectation of this new random variable.
Here are two examples, the first involves two independent random variables, and the second involves two dependent random variables, to illustrate the important difference.
Let $X$ be the number of people that are born on a given day and let $Y$ be the probability that a newborn child is a girl, viewed as a random quantity. It's probably reasonable to assume that Y is independent of X as if more people are born one day, that probably doesn't affect the distribution of girls and boys that are born on that day. Then $Z = XY$ is the number of girls born on a certain day. In this case, since the random variables are independent, we can just multiply their expectations:
$$E(Z) = E(XY) = E(X)E(Y)$$
Now let $X$ be the number of people in a household and $Y$ be the income person in that household. Then $Z =XY$ is the amount of income the whole household makes. We have, however, that $X$ and $Y$ are probably not independent, as if $X = 1$ then then we have a single-earner household and the income per person is probably higher that if we have $X = 5$, which probably means that we are looking at a family in which the children likely don't earn anything. Thus, in this situation
$$E(Z) = E(XY) \neq E(X)E(Y)$$
