Problem

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

$$F(x) = \begin{cases} 0, & x < 2, \\ \dfrac{(x - 2)^2}{12}, & 2 \leq x < 4, \\ 1 - \dfrac{(8 - x)^2}{24}, & 4 \leq x \leq 8, \\ 1, & x > 8. \end{cases}$$

(a) Sketch the graph of the probability density function of $X$.

(b) Find $E(X)$.

(c) Find the exact value of the interquartile range of $X$.