If X is Uniform(a,b), what is Var(X)?

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Multiple Choice

If X is Uniform(a,b), what is Var(X)?

Explanation:
Variance for a uniform distribution on an interval reflects how far, on average, values can be from the center, and it scales with the square of the interval length. For X ~ Uniform(a,b), you can think of X as a + (b−a)U where U ~ Uniform(0,1). Shifting by a does not change variance, and scaling by (b−a) scales the variance by (b−a)². Since Var(U) = 1/12 for a Uniform(0,1), we get Var(X) = (b−a)² × (1/12) = (b−a)²/12. A quick check using moments confirms this: E[X] = (a+b)/2 and E[X²] = (a² + ab + b²)/3, so Var(X) = E[X²] − E[X]² = (b−a)²/12. Thus the variance is (b−a)²/12.

Variance for a uniform distribution on an interval reflects how far, on average, values can be from the center, and it scales with the square of the interval length. For X ~ Uniform(a,b), you can think of X as a + (b−a)U where U ~ Uniform(0,1). Shifting by a does not change variance, and scaling by (b−a) scales the variance by (b−a)². Since Var(U) = 1/12 for a Uniform(0,1), we get Var(X) = (b−a)² × (1/12) = (b−a)²/12.

A quick check using moments confirms this: E[X] = (a+b)/2 and E[X²] = (a² + ab + b²)/3, so Var(X) = E[X²] − E[X]² = (b−a)²/12.

Thus the variance is (b−a)²/12.

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