9709 P63 - Jun 2021 - Q6 - 13 marks
10331
Alethia models the length of time, in minutes, by which her train is late on any day by the random variable \(X\) with probability density function given by \[f(x)=\begin{cases}\frac3{8000}(x-20)^2,&0\le x\le20,\\0,&\text{otherwise}.\end{cases}\]
(a) Find the probability that the train is more than 10 minutes late on each of two randomly chosen days. [4]
(b) Find \(E(X)\). [4]
(c) The median of \(X\) is denoted by \(m\).
(c) Show that \(m\) satisfies the equation \((m-20)^3=-4000\), and hence find \(m\) correct to 3 significant figures. [4]
(d) State one way in which Alethia’s model may be unrealistic. [1]
