February/March 2022 Paper 6 Probability and Statistics 2 Question 6

6. In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres,

travelled by the ball is modelled by the random variable X with probability density function 

f(x)=−k(x − 1)(x − 3) for 1 ≤ x ≤ 3, 0 otherwise

where k is a constant.

(a) Without calculation, explain why E(X) = 2.

(b) Show that k = 3/4

(c) Find Var(X)

One turn consists of rolling the ball 3 times and noting the largest value of X obtained. If this largest

value is greater than 2.5, the player scores a point.

(d) Find the probability that on a particular turn the player scores a point.