Suppose that the expectation of a random variable X is 5. Which of the following statements is true? A. There is a sample point at which X has the value 5. B. There is a sample point at which X has value greater than 5. C. There is a sample point at which X has a value greater than or equal to 5. D. None of the above.

GATE 1999 · Engineering Mathematics · Expectation · easy

Answer: C. There is a sample point at which X has a value greater than or equal to 5.

1. Recall the key property of expectation: If every x_i < 5, then E[X] = sum x_i * P(X = x_i) < 5 * sum P(X = x_i) = 5. Contradiction.
2. Rule out options A and B: E[X] = 3*0.5 + 7*0.5 = 5. X never equals 5 (rules out A). For B: if X = 5 always, E[X]=5 but no X > 5 (rules out B).