Part-discrete part-continuous distributionsAQA A-Level Further Maths: Subtopic test

1. 1
   The time $X$ minutes that a driver waits at a crossing is modelled as follows. With probability $0.4$ the light is green and $X=0$. Otherwise $X$ is continuous with probability density function $f(x)=kx$ for $0<x\le2$, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$0.3$
   • B$0.6$
   • C$0.15$
   • D$0.5$

   (b)

   Find $P(X\le1)$.

   [1 mark]

   • A$0.15$
   • B$0.55$
   • C$0.7$
   • D$0.3$

   (c)

   Find $P(X>1)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The daily rainfall $R$ mm at a weather station is modelled as follows. $P(R=0)=0.6$. On the other days $R$ is continuous with probability density function $f(r)=0.05r$ for $0<r\le4$.

   (a)

   Find $P(R<2)$.

   [1 mark]

   • A$0.1$
   • B$0.6$
   • C$0.7$
   • D$0.8$

   (b)

   Find $E(R)$.

   [1 mark]

   • A$\frac83$
   • B$0.4$
   • C$2$
   • D$\frac{16}{15}$

   (c)

   Find the probability that the rainfall on a given day exceeds $3$ mm.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A component fails immediately with probability $0.1$, so its lifetime $X$ years satisfies $P(X=0)=0.1$. Otherwise $X$ is continuous with probability density function $f(x)=k(6-x)$ for $0<x\le6$, where $k$ is a constant.

   (a)

   Show that $k=\frac1{20}$.

   [3 marks]

   (b)

   Find the median lifetime.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The claim $X$ (in hundreds of pounds) made by a policyholder in a year satisfies $P(X=0)=0.7$. When a claim is made, $X$ is continuous with probability density function $f(x)=kx^2$ for $0<x\le3$, where $k$ is a constant.

   (a)

   (i) Show that $k=\frac1{30}$.
   (ii) Find $P(X<1)$.
   (iii) Find $P(1<X<2)$.

   [6 marks]

   (b)

   Find the median and the upper quartile of $X$.

   [6 marks]

   Total for question 4: 12 marks

End of questions