Cumulative distribution functionsAQA A-Level Further Maths: Subtopic test

1. 1
   The continuous random variable $X$ has cumulative distribution function $F(x)=0$ for $x<0$, $F(x)=\frac{3x^2-x^3}{4}$ for $0\le x\le2$, and $F(x)=1$ for $x>2$.

   (a)

   Find $P(X\le1)$.

   [1 mark]

   • A$\frac34$
   • B$1$
   • C$\frac12$
   • D$2$

   (b)

   Find $P(X>1.5)$.

   [1 mark]

   • A$\frac{9}{16}$
   • B$\frac{27}{32}$
   • C$\frac12$
   • D$\frac{5}{32}$

   (c)

   Find the probability density function $f(x)$ of $X$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $X$ has probability density function $f(x)=\frac{3}{x^4}$ for $x\ge1$, and $f(x)=0$ otherwise.

   (a)

   Find $F(x)$ for $x\ge1$.

   [1 mark]

   • A$x^{-3}$
   • B$1-x^{-3}$
   • C$-12x^{-5}$
   • D$1-3x^{-3}$

   (b)

   Find $P(X>2)$.

   [1 mark]

   • A$\frac18$
   • B$\frac78$
   • C$\frac{3}{16}$
   • D$\frac14$

   (c)

   Find the median of $X$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has cumulative distribution function $F(x)=0$ for $x<0$, $F(x)=k(x^2+2x)$ for $0\le x\le2$, and $F(x)=1$ for $x>2$, where $k$ is a constant.

   (a)

   Find the value of $k$.

   [3 marks]

   (b)

   Find $f(x)$, and hence or otherwise find $P(0.5<X<1.5)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The time $X$ hours for which a customer waits has cumulative distribution function $F(x)=0$ for $x<0$, $F(x)=\frac{x^2}{6}$ for $0\le x\le2$, $F(x)=\frac{x}{3}$ for $2<x\le3$, and $F(x)=1$ for $x>3$.

   (a)

   (i) Find $P(1<X<2.5)$.
   (ii) Find the median of $X$.
   (iii) Find the probability density function $f(x)$.

   [6 marks]

   (b)

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

   [6 marks]

   Total for question 4: 12 marks

End of questions