Mean, variance, mode, median and quartilesEdexcel International A Level Further Maths: Subtopic test

1. 1
   The continuous random variable $X$ has probability density function $f(x)=\frac{3x^2}{8}$ for $0\le x\le2$, and $f(x)=0$ otherwise.

   (a)

   Find $E(X)$.

   [1 mark]

   • A$1$
   • B$\frac34$
   • C$\frac32$
   • D$\frac{12}{5}$

   (b)

   Find $\text{Var}(X)$.

   [1 mark]

   • A$\frac{3}{20}$
   • B$\frac{12}{5}$
   • C$\frac94$
   • D$\frac{9}{10}$

   (c)

   Find the probability that $X$ is greater than its mean.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The continuous random variable $Y$ has probability density function $f(y)=\frac y2$ for $0\le y\le2$, and $f(y)=0$ otherwise.

   (a)

   Find the median of $Y$.

   [1 mark]

   • A$1$
   • B$\frac43$
   • C$\sqrt3$
   • D$\sqrt2$

   (b)

   Find the interquartile range of $Y$.

   [1 mark]

   • A$1$
   • B$\sqrt3-1$
   • C$\sqrt3$
   • D$\sqrt3+1$

   (c)

   Write down the mode of $Y$, giving a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The continuous random variable $X$ has probability density function $f(x)=12x^2(1-x)$ for $0\le x\le1$, and $f(x)=0$ otherwise.

   (a)

   Find the mode of $X$.

   [3 marks]

   (b)

   Find $E(X)$ and $\text{Var}(X)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The time, $T$ minutes, taken to serve a customer at a till has probability density function $f(t)=\frac{3t^2}{64}$ for $0\le t\le4$, and $f(t)=0$ otherwise.

   (a)

   (i) Find the cumulative distribution function $F(t)$ for $0\le t\le4$.
   (ii) Find the median time.
   (iii) Find the interquartile range.

   [6 marks]

   (b)

   (i) Find $E(T)$.
   (ii) Find $\text{Var}(T)$.
   (iii) A manager says that half of all customers are served in less than the mean time. Use the model to evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions