Probability density functionsAQA 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$\frac19$
   • C$\frac1{27}$
   • D$9$

   (b)

   Find $P(X>2)$.

   [1 mark]

   • A$\frac{8}{27}$
   • B$\frac49$
   • C$\frac59$
   • D$\frac{19}{27}$

   (c)

   Find $P(1<X<2)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $X$ has probability density function $f(x)=\frac{x}{18}$ for $0\le x\le6$, and $f(x)=0$ otherwise.

   (a)

   Find $P(X<3)$.

   [1 mark]

   • A$\frac14$
   • B$\frac12$
   • C$\frac16$
   • D$\frac1{12}$

   (b)

   Find $P(X=4)$.

   [1 mark]

   • A$\frac29$
   • B$\frac49$
   • C$0$
   • D$\frac19$

   (c)

   Find the median of $X$.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

   Show that $k=\frac29$.

   [3 marks]

   (b)

   Find $P(1<X<2)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The time $X$ hours that a machine runs before a fault occurs has probability density function $f(x)=\frac{x}{4}$ for $0\le x<2$, $f(x)=\frac{4-x}{4}$ for $2\le x\le4$, and $f(x)=0$ otherwise.

   (a)

   (i) Show that $f$ is a valid probability density function.
   (ii) State the median of $X$, with a reason.
   (iii) Find the lower quartile of $X$.

   [6 marks]

   (b)

   Find the upper quartile of $X$, and hence the interquartile range.

   [6 marks]

   Total for question 4: 12 marks

End of questions