Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
The waiting time, $T$ minutes, for a passenger at a particular bus stop is represented by the probability density function $f(t) = \begin{cases} \frac{3}{4000}(20t - t^2), & 0 \leq t \leq 20, \\ 0, & \text{otherwise.} \end{cases}$
(a)[1]
Sketch the curve $y = f(t)$.
(b)[1]
Hence explain, without any calculation, why $\mathrm{E}(T) = 10$.
(c)[3]
Calculate $\mathrm{Var}(T)$.
(d)[2]
If $P(T < 10 + a) = p$, where $0 \le a \le 10$, find $P(10 - a < T < 10 + a)$ in terms of $p$.
(e)[3]
Calculate $P(8 < T < 12)$.
(f)[1]
State one reason why this model might not be realistic.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Parabola with tails pointing down, drawn from $x=0$ to $20$” …