Problem

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

$f(x) = \begin{cases} \dfrac{1}{4}(3 - x) & 1 \leqslant x \leqslant 2 \\[6pt] \dfrac{1}{4} & 2 < x \leqslant 3 \\[6pt] \dfrac{1}{4}(x - 2) & 3 < x \leqslant 4 \\[6pt] 0 & \text{otherwise} \end{cases}$

The cumulative distribution function of $X$ is $F(x)$.

(a) Show that $F(x) = \dfrac{1}{4}\left(-\dfrac{x^2}{2} + 3x\right) - \dfrac{5}{8}$ for $1 \leqslant x \leqslant 2$

(b) Find $F(x)$ for all values of $x$

(c) Find $P(1.2 < X < 3.1)$

(Total for Question 5 is 9 marks)