what is the rule for E(XY) given E(X) and E(Y)

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 \( Y \).

Would you like to explore this with more details or examples?

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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.

What is the rule for E(XY) given E(X) and E(Y)?

Explore the rule for finding the expected value of the product of two random variables, E(XY), given E(X) and E(Y). This page covers both independent and dependent cases, explaining the key formulas and concepts such as joint distributions and independence.

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