Problem

The continuous random variable $X$ has cumulative distribution function $F(x)$ given by

$F(x) = \begin{cases} 0 & x < 1 \\ \dfrac{1}{24}(x - 1)^2 & 1 \leqslant x \leqslant 3 \\ \dfrac{1}{6}(x - 2) & 3 < x \leqslant 6 \\ -k(x^2 - 20x + 52) & 6 < x \leqslant 10 \\ 1 & x > 10 \end{cases}$

where $k$ is a positive constant.

(a) Show that $k = \dfrac{1}{48}$

(b) Write down the value of $P(X = 4)$

(c) Find the value of $P(2.5 < X < 4.5) + P(5.5 < X < 8.5)$

The probability density function of $X$ is given by

$f(x) = \begin{cases} r(x) & 1 \leqslant x \leqslant 3 \\ \dfrac{1}{6} & 3 < x \leqslant 6 \\ s(x) & 6 < x \leqslant 10 \\ 0 & \text{otherwise} \end{cases}$

(d) Find $r(x)$ and $s(x)$

(Total for Question 4 is 9 marks)