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.
