Problem

A continuous random variable $X$ has probability density function $f$ given by

$$f(x) = \begin{cases} kx, & 0 \leq x < 1, \\ k(8 - x), & 1 \leq x \leq 8, \\ 0, & \text{otherwise,} \end{cases}$$

where $k$ is a constant.

(a) Show that $k = \frac{1}{25}$.

(b) Find the median value of $X$.

The random variable $Y$ is defined by $Y = \sqrt[3]{X}$.

(c) Find the probability density function of $Y$.