Rectangular distributionAQA A-Level Further Maths: Subtopic test

1. 1
   The time, $X$ minutes, that a customer waits for a lift is modelled by a continuous rectangular distribution on the interval $[2, 10]$.

   (a)

   What is the value of $f(x)$ for $2\le x\le 10$?

   [1 mark]

   • A$\frac{1}{10}$
   • B$\frac18$
   • C$\frac{1}{12}$
   • D$\frac12$

   (b)

   Find $\mathrm{P}(X>7)$.

   [1 mark]

   • A$\frac58$
   • B$\frac{3}{10}$
   • C$\frac38$
   • D$\frac78$

   (c)

   Find $\mathrm{P}(3<X<5.5)$, giving your answer as a fraction.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A length is measured to the nearest centimetre. The rounding error, $E$ cm, is modelled by a continuous rectangular distribution on the interval $[-0.5, 0.5]$.

   (a)

   What is $\mathrm{Var}(E)$?

   [1 mark]

   • A$\frac{1}{\sqrt{12}}$
   • B$\frac14$
   • C$0$
   • D$\frac{1}{12}$

   (b)

   Find $\mathrm{P}(E>0.3)$.

   [1 mark]

   • A$0.2$
   • B$0.3$
   • C$0.8$
   • D$0.4$

   (c)

   Find the probability that the rounding error is more than $0.4$ cm in size, that is $\mathrm{P}(|E|>0.4)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has a rectangular distribution on the interval $[a, b]$, where $a<b$, so that $f(x)=\frac{1}{b-a}$ for $a\le x\le b$ and $f(x)=0$ otherwise.

   (a)

   Prove that $\mathrm{E}(X)=\frac{a+b}{2}$.

   [3 marks]

   (b)

   Prove that $\mathrm{Var}(X)=\frac{(b-a)^2}{12}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Trains leave a station every 15 minutes. A passenger who has not looked at the timetable arrives at a random time. The waiting time, $T$ minutes, for the next train is modelled by a continuous rectangular distribution on the interval $[0, 15]$.

   (a)

   (i) Find $\mathrm{E}(T)$ and the standard deviation of $T$.
   (ii) Find the probability that $T$ is within one standard deviation of its mean.

   [6 marks]

   (b)

   (i) Three independent passengers each arrive at random. Find the probability that exactly two of them wait more than 10 minutes.
   (ii) Evaluate whether the rectangular model is suitable for passengers who regularly commute and know the timetable.

   [6 marks]

   Total for question 4: 12 marks

End of questions