Final answer:
To compute E(XY), you multiply each joint outcome by its probability and sum these products. E(X) and E(Y) are calculated separately and then compared to E(XY) to check for independence. If E(XY) equals E(X)E(Y), X and Y are independent.
Step-by-step explanation:
To compute the expected value of the product of two random variables X and Y, denoted as E(XY), you need to multiply each joint outcome (xy) by its probability P(X=x & Y=y) and sum all of these products together. The formula to calculate E(XY) is Σ (xy) * P(X=x & Y=y).
To find E(X) and E(Y), compute the expectation of each random variable separately using Σ xP(x) for E(X) and Σ yP(y) for E(Y), then multiply them together to obtain E(X)E(Y). This value is compared to E(XY) to see if the variables are independent. If E(XY) = E(X)E(Y), then X and Y are independent.
Comparing E(XY) and E(X)E(Y) allows you to evaluate the correlation between X and Y. If they have a non-zero correlation, then their expected product will differ from the product of their marginal expectations.