If x and y are two random variables, what is E[xy]?

Question: If x and y are two random variables, what is E[xy]?

If $x$ and $y$ are two random variables, the expected value of their product $xy$ is given by:

$E[xy] = \sum_{i}\sum_{j} x_iy_j P(X=x_i, Y=y_j)$

In continuous cases, where $x$ and $y$ are continuous random variables, this is expressed as an integral:

$E[xy] = \int\int xy f(x, y) \,dx\,dy$

where $f(x, y)$ is the joint probability density function of $x$ and $y$.

In simpler terms, the expected value of the product of two random variables is found by multiplying each pair of possible values of $x$ and $y$ by the probability of that particular combination occurring, and then summing or integrating over all possible pairs.

It's important to note that if $x$ and $y$ are independent, then $E[xy] = E[x] \cdot E[y]$. If they are not independent, the expected value of the product is influenced by the covariance between $x$ and $y$, and it can be expressed as:

$E[xy] = E[x] \cdot E[y] + \text{cov}(x, y)$

where $\text{cov}(x, y)$ is the covariance between $x$ and $y$.