How is the expected value E[XY] calculated in probability theory?

How is the expected value E[XY] calculated in probability theory?

• A. As the product of variances of X and Y
• B. As the sum of expected values of X and Y
• C. As E[X]E[Y] if X and Y are independent
• D. As E[X + Y]

Answer: (C)

The expected value E[XY] for two random variables X and Y can be calculated as E[X]E[Y] if X and Y are independent. This property stems from the fundamental nature of expectation in probability theory. When two variables are independent, the occurrence of one does not affect the occurrence of the other, leading to a situation where the joint expectation can indeed be expressed as the product of their individual expectations. For independent variables, the calculation of E[XY] becomes simpler because it follows the rules of multiplication of expectations, allowing us to derive E[XY] without needing a joint distribution. This applies broadly in probability theory, making it a highly useful property for calculations involving independent random variables. In contrast, the other options do not accurately describe the relationship between expected values of random variables. Calculating E[XY] as the product of their variances, for instance, is neither valid nor reflective of how expectation operates. Similarly, summing the expected values E[X] and E[Y] does not yield E[XY], as this operation pertains to linear combinations rather than products. Finally, while E[X + Y] indeed represents the expected value for the sum of two random variables, it does not relate to E[XY] in the

The expected value E[XY] for two random variables X and Y can be calculated as E[X]E[Y] if X and Y are independent. This property stems from the fundamental nature of expectation in probability theory. When two variables are independent, the occurrence of one does not affect the occurrence of the other, leading to a situation where the joint expectation can indeed be expressed as the product of their individual expectations.

For independent variables, the calculation of E[XY] becomes simpler because it follows the rules of multiplication of expectations, allowing us to derive E[XY] without needing a joint distribution. This applies broadly in probability theory, making it a highly useful property for calculations involving independent random variables.

In contrast, the other options do not accurately describe the relationship between expected values of random variables. Calculating E[XY] as the product of their variances, for instance, is neither valid nor reflective of how expectation operates. Similarly, summing the expected values E[X] and E[Y] does not yield E[XY], as this operation pertains to linear combinations rather than products. Finally, while E[X + Y] indeed represents the expected value for the sum of two random variables, it does not relate to E[XY] in the