Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.

Two numbers being selected from 1,2,3,4,5 in 5×4=20 ways. Let P(X) denote the probability of X being the larger number.
∴P(2) = 2/20 (cases (1,2) and (2,1))
∴P(3) = 4/20 (cases (1,3), (3,1), (3,2) and (2,3))
∴P(4) = 6/20 (cases (1,4), (4,1), (4,2), (2,4), (3,4) and (4,3))
∴P(5) = 8/20 (cases (1,5), (5,1), (5,2), (2,5), (3,5), (5,3), (5,4) and (4,5))
Now, mean = Σᵢ₌₂⁵ X × P(X) = 4
variance = Σᵢ₌₂⁵ X² × P(X) − (Σᵢ₌₂⁵ X × P(X))² = 17/4 = 1
These are the required answers.