Let X have a uniform distribution on the interval (2, 4). Find the probability that the sum of 2 independent observations of X is greater than 7.

Question

Let X have a uniform distribution on the interval (2, 4). Find the probability that the sum of 2 independent observations of X is greater than 7.

Answer 1

You can solve this by using a little geometry.

Picture a box [2,4] x [2,4] with density 1/2 x 1/2 = 1/4

X >= 7 consists of the triangle at vertices [3,4], [4, 4], [4,3]

Thus, the area of the triangle is 1/2 * 1 * 1 = 1/2 and, multiplying by the density 1/4, we get 1/2 * 1/4 = 1/8.