Which statement expresses independence between two random variables X and Y?

• A. P(X,Y) = P(X) + P(Y).
• B. P(X,Y) > P(X)P(Y).
• C. X and Y have the same distribution.
• D. P(X,Y) = P(X)P(Y).

Answer: (D)

Independence means the outcome of one variable tells us nothing about the other, so their joint behavior is just the product of their individual behaviors. For discrete variables, this is written as P(X=x, Y=y) = P(X=x) P(Y=y) for every pair of values x and y. That exact equality captures the idea that knowing X provides no information about Y, and vice versa. If this equality holds across all possible values, the variables are independent; if it fails for some pair, there is some dependence between them.

The other statements don’t express independence. Equating the joint probability to a sum of marginals would not generally hold and would clash with how probabilities combine. Having the same distribution means the marginals X and Y share the same shape, but independence is about how the two variables interact, not just their individual distributions. And a joint probability larger than the product suggests a positive association, which indicates dependence rather than independence.