Problem

A continuous random variable $X$ has cumulative distribution function

$F(x) = \begin{cases} 0 & x < 2 \\ \dfrac{1}{20}(x^2 - 4) & 2 \leqslant x \leqslant 4 \\ \dfrac{1}{5}(2x - 5) & 4 < x \leqslant 5 \\ 1 & x > 5 \end{cases}$

(a) Calculate $P(X > 4)$

(b) Find the value of $a$ such that $P(3 < X < a) = 0.642$

(c) Find the probability density function of $X$, specifying it for all values of $x$.

(Total for Question 1 is 10 marks)