Which expression equals Var(X) for a discrete random variable X?

• A. Var(X) = sum (x - μ)^2.
• B. Var(X) = E[(X - E[X])^2].
• C. Var(X) = E[X^2] - E[X].
• D. Var(X) = sum P(X=x).

Answer: (B)

Variance measures how spread out X is around its average. It is defined as the expected squared distance from the mean: Var(X) = E[(X - μ)^2], where μ = E[X]. For a discrete variable this is computed by weighting the squared deviations by their probabilities: Var(X) = sum over all values x of (x - μ)^2 P(X = x). This form directly captures both how far outcomes are from the mean and how likely those outcomes are.

This is why the expression above is the correct one. An equivalent way to write it is Var(X) = E[X^2] - (E[X])^2, which is the same thing in different form. The option that uses E[X^2] - E[X] would miss squaring the mean and thus is not correct. The option that sums P(X = x) without any deviation term would just equal 1, not a measure of spread.