Problem

The continuous random variable $X$ has probability density function given by

$f(x) = \begin{cases} ax & 0 \leqslant x \leqslant 4 \\ bx + c & 4 < x \leqslant 8 \\ 0 & \text{otherwise} \end{cases}$

where $a$, $b$ and $c$ are constants.

Figure 1 shows the graph of the probability density function $f(x)$.

The graph consists of two straight line segments of equal length joined at the point where $x = 4$.

(a) Show that $a = \dfrac{1}{16}$

(b) Hence find

(i) the value of $b$

(ii) the value of $c$

(c) Using algebraic integration, show that $\text{Var}(X) = \dfrac{8}{3}$

(d) Find, to 2 decimal places, the lower quartile and the upper quartile of $X$

A statistician claims that

$P(-\sigma < X - \mu < \sigma) > 0.5$

where $\mu$ and $\sigma$ are the mean and standard deviation of $X$.

(e) Show that the statistician’s claim is correct.

(Total for Question 7 is 15 marks)