The rule for $E(XY)$, the expected value of the product of two random variables $X$ and $Y$, depends on whether $X$ and $Y$ are independent or not.

1. If $X$ and $Y$ are independent: $E(XY) = E(X) \cdot E(Y)$ This result holds because, for independent random variables, the expectation of their product equals the product of their expectations.

2. If $X$ and $Y$ are not independent: There is no simplified general formula for $E(XY)$, and it will depend on the joint distribution of $X$ and $Y$. In such cases, you would need to use: $E(XY) = \int \int xy \, f_{XY}(x, y) \, dx \, dy$ where $f_{XY}(x, y)$ is the joint probability density function of $X$ and $Y$.

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Here are 5 related questions for further learning:

1. What is the covariance of $X$ and $Y$, and how is it related to $E(XY)$?
2. How do you calculate $E(XY)$ when $X$ and $Y$ are dependent?
3. What is the variance of the product of two random variables, $\text{Var}(XY)$?
4. How does the law of total expectation apply to $E(XY)$?
5. Can you explain the concept of joint probability distributions?

Tip: Independence of random variables greatly simplifies the calculation of their combined expectations.