Exponential distribution model, pdf and CDFAQA A-Level Further Maths: Subtopic test

1. 1
   Calls to a helpline arrive at random, independently of each other, at a constant average rate of 3 per hour. The time, $X$ hours, between successive calls is modelled by an exponential distribution with probability density function $f(x)=3e^{-3x}$ for $x\ge0$.

   (a)

   Find the probability that the time between two successive calls is less than half an hour.

   [1 mark]

   • A$0.777$
   • B$0.223$
   • C$0.669$
   • D$0.950$

   (b)

   Find the probability that the time between two successive calls is more than one hour.

   [1 mark]

   • A$0.950$
   • B$0.0498$
   • C$0.149$
   • D$0.368$

   (c)

   Find the probability that the time between two successive calls is between 12 minutes and 30 minutes.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The lifetime, $T$ years, of a certain type of component is modelled by a continuous random variable with cumulative distribution function $F(t)=1-e^{-0.2t}$ for $t\ge0$.

   (a)

   Find the probability density function of $T$.

   [1 mark]

   • A$-0.2e^{-0.2t}$
   • B$e^{-0.2t}$
   • C$0.2e^{-0.2t}$
   • D$-5e^{-0.2t}$

   (b)

   Find the probability that a component lasts more than 8 years.

   [1 mark]

   • A$0.798$
   • B$0.819$
   • C$0.0404$
   • D$0.202$

   (c)

   The pdf of $T$ is $f(t)=0.2e^{-0.2t}$ for $t\ge0$. Use integration to find the exact probability that a component fails within 5 years.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time, $X$ minutes, between successive arrivals at a supermarket checkout is modelled by an exponential distribution with parameter $\lambda$, where $\lambda>0$ is a constant. It is known that $\mathrm{P}(X>2)=0.3$.

   (a)

   Find the value of $\lambda$.

   [3 marks]

   (b)

   Find the time $t$ such that $90\%$ of the gaps between arrivals are shorter than $t$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Breakdowns of a machine occur at random, independently of each other, at a constant average rate. The time, $X$ weeks, between successive breakdowns is modelled by the probability density function $f(x)=0.25e^{-0.25x}$ for $x\ge0$.

   (a)

   (i) Show, by integration, that the cumulative distribution function of $X$ is $F(x)=1-e^{-0.25x}$.
   (ii) Find the probability that the time between two successive breakdowns is between 2 and 6 weeks.

   [6 marks]

   (b)

   (i) State two conditions that must hold for the exponential model to be appropriate.
   (ii) Assuming the model, find the probability that three successive times between breakdowns are each longer than 6 weeks.
   (iii) The engineer notices that breakdowns tend to occur in clusters shortly after each service. Evaluate whether the model, and your answer to (ii), are reliable.

   [6 marks]

   Total for question 4: 12 marks

End of questions