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A variable X takes values x_i such that a <= x_i <= b for i = 1, 2,..., n. Which of the following is always true about Var(X)?

1. a²/4 <= Var(X)
2. (b - a)² >= Var(X)
3. a <= Var(X) <= b
4. a² <= Var(X) <= b²

Correct answer: (b - a)² >= Var(X)

Solution

By Popoviciu's inequality, if a <= X <= b, then Var(X) <= (b-a)²/4. Since (b-a)²/4 <= (b-a)², it follows that Var(X) <= (b-a)². This makes option B always true.

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