Further Statistics 2: Exponential distributionAQA A-Level Further Maths: Topic test

1. 1
   The time, $X$ hours, between consecutive tremors recorded at a seismic station is modelled by an exponential distribution with parameter $\lambda=0.2$.

   (a)

   Find $P(X>5)$.

   [1 mark]

   • A$0.3679$
   • B$0.6321$
   • C$0.8187$
   • D$0.0067$

   (b)

   Find $E(X)$.

   [1 mark]

   • A$0.2$
   • B$25$
   • C$5$
   • D$3.47$

   (c)

   Find $P(2<X<6)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Emails arrive in an inbox at random, independently of each other, at a constant average rate of $4$ per hour. Let $T$ hours be the time between two consecutive emails.

   (a)

   Which is the distribution of $T$?

   [1 mark]

   • AAn exponential distribution with parameter $15$
   • BAn exponential distribution with parameter $4$
   • CA Poisson distribution with mean $4$
   • DAn exponential distribution with parameter $0.25$

   (b)

   Find the mean time between consecutive emails, in minutes.

   [1 mark]

   • A$4$
   • B$0.25$
   • C$240$
   • D$15$

   (c)

   Find the probability that the time between two consecutive emails is more than half an hour.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time, $X$ hours, that a patient waits in an emergency department is modelled by an exponential distribution with parameter $\lambda$. The probability that a patient waits more than $3$ hours is $0.3$.

   (a)

   Find the value of $\lambda$.

   [3 marks]

   (b)

   Find the median waiting time and the probability that a patient waits between $1$ and $2$ hours.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Customers arrive at a bike-hire dock at random, independently of each other, at a constant average rate of $5$ per hour, so the arrivals form a Poisson process. Let $T$ hours be the time between consecutive arrivals.

   (a)

   Show that the cumulative distribution function of $T$ is $F(t)=1-e^{-5t}$ for $t\ge0$. Hence find the probability density function of $T$ and $E(T)$.

   [6 marks]

   (b)

   Find the standard deviation of $T$ in minutes. Find the probability that the time between two consecutive arrivals is less than $6$ minutes, and the probability that it is longer than the mean time.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The distance, $X$ km, between consecutive potholes along a rural road is modelled by an exponential distribution with mean $2.5$ km.

   (a)

   Find the standard deviation of $X$.

   [1 mark]

   • A$6.25$ km
   • B$1.58$ km
   • C$0.4$ km
   • D$2.5$ km

   (b)

   Find $P(X<1)$.

   [1 mark]

   • A$0.6703$
   • B$0.3297$
   • C$0.6321$
   • D$0.4$

   (c)

   Find the distance $d$ such that $P(X>d)=0.05$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Lightning strikes at a mountain observatory occur at random, independently of each other, at a constant average rate of $1.5$ per hour. Let $M$ minutes be the time between two consecutive strikes.

   (a)

   Find the mean time between consecutive strikes.

   [1 mark]

   • A$90$ minutes
   • B$1.5$ hours
   • C$40$ minutes
   • D$0.67$ minutes

   (b)

   Which condition must hold for an exponential distribution to be a suitable model for the time between strikes?

   [1 mark]

   • AStrikes occur singly, independently of each other and at a constant average rate.
   • BStrikes occur at regular intervals of equal length.
   • CThe rate of strikes increases during a storm.
   • DThe number of strikes in each hour is always $1.5$.

   (c)

   Find the probability that the time between two consecutive strikes is more than one hour.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The lifetime, $X$ years, of a type of server fan is modelled by an exponential distribution with mean $5$ years. A data centre uses $10$ of these fans, whose lifetimes are independent of each other.

   (a)

   Find the standard deviation of $X$ and the probability that a fan lasts more than $8$ years.

   [3 marks]

   (b)

   Find the probability that exactly $2$ of the $10$ fans last more than $8$ years. State an assumption that your calculation needs.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two models are described. Model 1: The lifetime, $T$ years, of a type of sensor is modelled by an exponential distribution with median $3$ years. Model 2: Accidents at a road junction occur at random, independently of each other, at a constant average rate of $2.5$ per month. Let $Y$ months be the time between consecutive accidents.

   (a)

   For Model 1, find the parameter $\lambda$, the mean and standard deviation of $T$, and $P(T>6)$.

   [6 marks]

   (b)

   For Model 2, state the distribution of $Y$ and find $P(Y>0.5)$. Starting from an accident, find the probability that there is no accident in the next $2$ months in two ways: using a Poisson distribution and using the distribution of $Y$.

   [6 marks]

   Total for question 8: 12 marks

End of questions