How is the expected value of a discrete random variable X with PMF p(x) computed?

• A. E[X] = Σ p(x).
• B. E[X] = Σ P(X=x).
• C. E[X] = Σ x P(X=x).
• D. E[X] = max{x: P(X=x) > 0}.

Answer: (C)

The expected value of a discrete random variable is the probability-weighted average of its possible values. If X can take values x with probabilities p(x), then the long-run average outcome is E[X] = sum over all x of x p(x). This weighting by p(x) is what centers the average around where the distribution puts its mass.

Why the other forms don’t fit: summing the probabilities without the value x gives the total probability, which is always 1, not the average outcome. It’s just a measure of how likely anything is to occur, not where the values cluster. taking the maximum value with positive probability only reflects the largest possible outcome, not its typical size. Thus the weighted sum x p(x) captures both the values and their likelihoods, yielding the correct expected value.