Further Statistics 2: Continuous probability distributionsEdexcel A-Level Further Maths: Topic test

1. 1
   The continuous random variable $X$ is the time, in hours, that a student spends on a task. It has probability density function $\mathrm{f}(x)=kx(3-x)$ for $0\leq x\leq3$, and $\mathrm{f}(x)=0$ otherwise, where $k$ is a constant.

   (a)

   What is the value of $k$?

   [1 mark]

   • A$\frac92$
   • B$\frac29$
   • C$\frac13$
   • D$\frac19$

   (b)

   What is $\mathrm{P}(X>2)$?

   [1 mark]

   • A$\frac{20}{27}$
   • B$\frac29$
   • C$\frac{7}{27}$
   • D$\frac49$

   (c)

   Find $\mathrm{E}(X)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $Y$ is the waiting time, in minutes, for a lift. It has cumulative distribution function $\mathrm{F}(y)=0$ for $y<0$, $\mathrm{F}(y)=\dfrac{4y-y^2}{4}$ for $0\leq y\leq2$, and $\mathrm{F}(y)=1$ for $y>2$.

   (a)

   What is the probability density function of $Y$ for $0\leq y\leq2$?

   [1 mark]

   • A$1-\dfrac y2$
   • B$\dfrac{4y-y^2}{4}$
   • C$\dfrac{4-y}{4}$
   • D$\dfrac{y^2}{4}-y$

   (b)

   What is $\mathrm{P}(0.5<Y<1.5)$?

   [1 mark]

   • A$0.9375$
   • B$0.4375$
   • C$0.0625$
   • D$0.5$

   (c)

   Find the median of $Y$, giving your answer to 3 decimal places.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time $T$, in minutes, that a customer waits for a bus is uniformly distributed over the interval $[0,12]$.

   (a)

   Write down $\mathrm{E}(T)$ and find $\mathrm{Var}(T)$.

   [3 marks]

   (b)

   Find the probability that $T$ is within one standard deviation of its mean.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The continuous random variable $X$ is the time, in hours, taken to complete a repair. It has probability density function $\mathrm{f}(x)=\dfrac x6$ for $0\leq x\leq2$, $\mathrm{f}(x)=\dfrac{6-x}{12}$ for $2<x\leq6$, and $\mathrm{f}(x)=0$ otherwise.

   (a)

   Find the cumulative distribution function $\mathrm{F}(x)$ for all values of $x$.

   [6 marks]

   (b)

   Find the median and the mean of $X$, and use them, together with the mode, to describe the skewness of the distribution.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A random number generator produces a value $R$ that is uniformly distributed over the interval $[-4,6]$.

   (a)

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

   [1 mark]

   • A$0$
   • B$5$
   • C$1$
   • D$2$

   (b)

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

   [1 mark]

   • A$\frac56$
   • B$\frac{25}{3}$
   • C$\frac{100}{3}$
   • D$\frac{10}{\sqrt{12}}$

   (c)

   Find $\mathrm{P}(R>2.5)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The continuous random variable $X$ is the time, in hours, before a battery next needs recharging. It has probability density function $\mathrm{f}(x)=\frac{3}{16}(4-x^2)$ for $0\leq x\leq2$, and $\mathrm{f}(x)=0$ otherwise.

   (a)

   What is $\mathrm{P}(X<1)$?

   [1 mark]

   • A$\frac{5}{16}$
   • B$\frac{3}{16}$
   • C$\frac12$
   • D$\frac{11}{16}$

   (b)

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

   [1 mark]

   • A$\frac34$
   • B$1$
   • C$\frac45$
   • D$\frac32$

   (c)

   Find $\mathrm{Var}(X)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A technician spends $X$ hours on a call-out, where $X$ has probability density function $\mathrm{f}(x)=\dfrac{x+1}{4}$ for $0\leq x\leq2$, and $\mathrm{f}(x)=0$ otherwise. The profit, in hundreds of pounds, from a call-out is $P=4X-X^2$.

   (a)

   Show that $\mathrm{E}(X)=\frac76$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A tram is due at a stop sometime after 09:00. The time $T$, in minutes after 09:00, at which it arrives is uniformly distributed over the interval $[0,10]$. A passenger arrives at the stop at 09:04.

   (a)

   Use integration to derive $\mathrm{E}(T)$ and $\mathrm{Var}(T)$.

   [6 marks]

   (b)

   The tram has not arrived when the passenger arrives. Find the probability that the passenger then waits for more than $3$ minutes, and find the expected further wait.

   [6 marks]

   Total for question 8: 12 marks

End of questions