I am trying to understand the value of $\bar{x_{1} x_{2}}+E(x_{1} x_{2})$. For all $i$, $E(x_{i})$ and $\sigma_{i}$ are given. Wikipedia gives the joint probability density function:
$E(XY) = \int \int x y j(x,y) dx dy$
then I can find out from wikipedia that:
$E(XY) = Cov(X,Y) + E(X)E(Y)$
and by Cauchy-Swartz:
$| Cov(X,Y) | \leq \sigma(X) \sigma(Y)$
but I cannot find a precise formula to find the value of $E(XY)$, only an upper bound with finite variances. A crux point to find the matrix $\rho$ so that I can calculate the $\sigma$ matrix. So how can I calculate the $E(XY)$ when only $E(x_{i})$ and $\sigma_{i}$ for all $i$ are given?
Best Answer
Such a formula cannot exist, since it would compute $E(XY)$ only using (characteristics associated to) the separate distributions of $X$ and $Y$. $E(XY)$ depends, as you noted, on the joint distribution of $X$ and $Y$.
The missing figure you need to get a formula as you would like is the correlation coefficient.