Which expression defines the expected value E[X] for a discrete random variable X?

• A. E[X] = sum P(X=x).
• B. E[X] = sum x.
• C. E[X] = max_x x P(X=x).
• D. E[X] = sum x P(X=x).

Answer: (D)

The expected value of a discrete random variable is the weighted average of its possible values, with each value weighted by its probability of occurring. This is computed as E[X] = sum over all x of x · P(X = x). Intuitively, if you could repeat the experiment many times and average the outcomes, the long-run average would approach this weighted mean, reflecting both how large the outcomes can be and how likely each outcome is.

If you only sum the probabilities, you’re just getting 1, which isn’t a value representing an average outcome. If you take the maximum of x · P(X = x), you’re picking a single outcome-weight product, not averaging across all possibilities. And summing x without including the probabilities ignores how likely each value is, so it doesn’t properly reflect the distribution unless all outcomes are equally likely.