Problem

A continuous random variable $X$ has cumulative distribution function

$F(x) = \begin{cases} 0 & x < 4 \\ k(x - 4) & 4 \leqslant x \leqslant 10 \\ 1 & x > 10 \end{cases}$

where $k$ is a constant.

(a) Show that $k = \dfrac{1}{6}$

(b) Write down the value of $P(X = 5)$

(c) Find $P(5 < X < 7)$

(d) Specify the probability density function of $X$, for all values of $x$

(e) Hence sketch the probability density function of $X$

(f) Write down the value of $E(X)$

(g) Find $E(2X^2 - 1)$

You must make your method clear.

(Total for Question 2 is 12 marks)