Let X have a uniform distribution on the interval (8, 13).

Question

Let X have a uniform distribution on the interval (8, 13). Find the probability that the sum of 2 independent observations of X is greater than 25.

Answer 1

It is great to solve this geometrically. X ~ U[8, 13] Y ~ U[8, 13]

The density of the 2 random variables is 1/5 * 1/5 = 1/25 on [8, 13] x [8,13]

X+Y >= 25 on the upper, right triangle with vertices (12, 13), (13, 12), (13, 13)

This triangle has area 1/2 * 1 * 1 = 1/2, so P(X + Y) >= 25 = 1/2 * 1/25 = 1/50.