Which statement best defines the expected value in a theoretical distribution?

Which statement best defines the expected value in a theoretical distribution?

• A. The most probable single outcome.
• B. The weighted average of all possible outcomes.
• C. The maximum probability of any outcome.
• D. The sum of probabilities equals one.

Answer: (B)

Expected value is the long-run average outcome you would expect if you could repeat the random process many times, with each outcome weighted by how likely it is. It is the weighted average of all possible outcomes, calculated by summing each outcome value times its probability. In a discrete distribution, that’s E[X] = Σ x · P(X = x); in a continuous distribution, it’s the integral of x with respect to the probability density function. This differs from picking the single most probable outcome, which is the mode, not the average. It also isn’t just the maximum probability itself, which tells you how likely the most likely outcome is but not the average you’d expect over many trials. And the fact that probabilities sum to one describes a property of the distribution as a whole, not the expected value itself. The expected value captures the center of gravity of the distribution, representing the long-run average outcome. For example, if outcomes are 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, the expected value is 1(0.2) + 2(0.5) + 3(0.3) = 2.1, which is the average you’d anticipate over many trials, even though you might never see exactly 2.1 in any single trial.

Expected value is the long-run average outcome you would expect if you could repeat the random process many times, with each outcome weighted by how likely it is. It is the weighted average of all possible outcomes, calculated by summing each outcome value times its probability. In a discrete distribution, that’s E[X] = Σ x · P(X = x); in a continuous distribution, it’s the integral of x with respect to the probability density function.

This differs from picking the single most probable outcome, which is the mode, not the average. It also isn’t just the maximum probability itself, which tells you how likely the most likely outcome is but not the average you’d expect over many trials. And the fact that probabilities sum to one describes a property of the distribution as a whole, not the expected value itself. The expected value captures the center of gravity of the distribution, representing the long-run average outcome. For example, if outcomes are 1, 2, and 3 with probabilities 0.2, 0.5, and 0.3, the expected value is 1(0.2) + 2(0.5) + 3(0.3) = 2.1, which is the average you’d anticipate over many trials, even though you might never see exactly 2.1 in any single trial.