Mean and variance, and link to Poisson processesAQA A-Level Further Maths: Subtopic test

1. 1
   The lifetime, $X$ days, of a type of battery is modelled by an exponential distribution with parameter $\lambda=0.05$.

   (a)

   Find the mean lifetime of a battery.

   [1 mark]

   • A$0.05$ days
   • B$20$ days
   • C$400$ days
   • D$0.0025$ days

   (b)

   Find the variance of the lifetime.

   [1 mark]

   • A$400$
   • B$20$
   • C$0.0025$
   • D$40$

   (c)

   Find the probability that a battery lasts longer than its mean lifetime.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Particles strike a detector at random, independently of each other, at a constant average rate of 12 per hour, so the number of strikes follows a Poisson process. Let $T$ minutes be the time between two successive strikes.

   (a)

   Which statement correctly describes $T$?

   [1 mark]

   • A$T$ has a Poisson distribution with mean 12
   • B$T$ has an exponential distribution with mean 12 minutes
   • C$T$ has a normal distribution with mean 5 minutes
   • D$T$ has an exponential distribution with mean 5 minutes

   (b)

   Find the probability that the time between two successive strikes is more than 10 minutes.

   [1 mark]

   • A$0.865$
   • B$0.368$
   • C$0.135$
   • D$0.0183$

   (c)

   Find the standard deviation of $T$ in minutes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has an exponential distribution with parameter $\lambda>0$, so that $f(x)=\lambda e^{-\lambda x}$ for $x\ge0$. You may use the fact that $xe^{-\lambda x}\to0$ and $x^2e^{-\lambda x}\to0$ as $x\to\infty$.

   (a)

   Prove that $\mathrm{E}(X)=\frac{1}{\lambda}$.

   [3 marks]

   (b)

   Prove that $\mathrm{Var}(X)=\frac{1}{\lambda^2}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Faults occur at random, independently of each other, along a long cable at a constant average rate of 0.8 per kilometre, so the number of faults in a length of cable follows a Poisson process. The distance, $X$ km, between successive faults is modelled by an exponential distribution.

   (a)

   (i) State the distribution of $X$, and find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.
   (ii) Find the probability that the distance between two successive faults is greater than 2 km.
   (iii) Explain why your answer to (ii) is also the probability that a given 2 km length of cable contains no faults.

   [6 marks]

   (b)

   A technician inspects the cable from one end. The distance to the first fault is also exponential with the same parameter.
   (i) Find the probability that the first fault is between 1 km and 3 km from the end.
   (ii) Find the probability that there are exactly 2 faults in the first 3 km.
   (iii) The technician finds that faults are more common near the joints of the cable. Evaluate the effect on your answers to (i) and (ii).

   [6 marks]

   Total for question 4: 12 marks

End of questions