Probability density and cumulative distribution functionsEdexcel International A Level Further Maths: Subtopic test

1. 1
   The continuous random variable $X$ has probability density function $f(x)=kx^2$ for $0\le x\le3$, and $f(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [1 mark]

   • A$\frac13$
   • B$\frac1{27}$
   • C$9$
   • D$\frac19$

   (b)

   Find $P(X>2)$.

   [1 mark]

   • A$\frac{8}{27}$
   • B$\frac{19}{27}$
   • C$\frac13$
   • D$\frac59$

   (c)

   Find the cumulative distribution function $F(x)$ for $0\le x\le3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $Y$ has cumulative distribution function $F(y)=0$ for $y<1$, $F(y)=\frac{y^2-1}{8}$ for $1\le y\le3$, and $F(y)=1$ for $y>3$.

   (a)

   Find $P(Y\le2)$.

   [1 mark]

   • A$\frac12$
   • B$\frac34$
   • C$\frac38$
   • D$\frac58$

   (b)

   Find the probability density function $f(y)$ for $1\le y\le3$.

   [1 mark]

   • A$\frac y4$
   • B$\frac{y}{8}$
   • C$\frac{y^3}{24}-\frac y8$
   • D$\frac{y^2-1}{8}$

   (c)

   Find $P(1.5<Y\le2.5)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has probability density function $f(x)=kx$ for $0\le x\le2$, $f(x)=k(6-2x)$ for $2<x\le3$, and $f(x)=0$ otherwise, where $k$ is a constant.

   (a)

   Show that $k=\frac13$.

   [3 marks]

   (b)

   Find the cumulative distribution function $F(x)$ for $2<x\le3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The time, $T$ hours, that a customer waits for a repair has cumulative distribution function $F(t)=0$ for $t<0$, $F(t)=\frac{4t^3-t^4}{27}$ for $0\le t\le3$, and $F(t)=1$ for $t>3$.

   (a)

   (i) Find the probability density function $f(t)$.
   (ii) Show that $f(t)\ge0$ for $0\le t\le3$.
   (iii) Find $P(1<T<2.5)$.

   [6 marks]

   (b)

   (i) Find the probability that a customer waits for more than 2 hours.
   (ii) Two customers' waiting times are independent. Find the probability that both wait for more than 2 hours.
   (iii) Given that a customer has already waited for more than 1 hour, find the probability that they wait for more than 2 hours.

   [6 marks]

   Total for question 4: 12 marks

End of questions