let $X$,$Y$ be independent Poisson distributed random variables with parameter $\alpha$ and $\beta$ respectively. $E(XY)$?

Question

$$E(XY) = \sum_{x,y} xy f(x,y),$$

but I don´t have the $f(x,y)$. $X+Y$ would be Poisson distributed with parameter $\alpha + \beta$, but what about $XY$? Not sure what else to do.

Thanks in advance.

Answer 1

As $X$ and $Y$ are independent variables, the joint PDF $f_{X,Y}=f_Xf_Y$ i.e. it is the product of the marginal PDFs.

Thus - \begin{align} E(XY)&=\int\int XYf_{X,Y}\\ &=\int\int XYf_Xf_Y\\ &=\int Xf_X \int Yf_Y\\ &=E(x)E(Y) \end{align} This holds for any independent $X$, $Y$, and not just Poisson variables. Similarly, this can be generalized to $n$ variables.