Problem

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

$$f(x) = \begin{cases} \dfrac{1}{6}\big(x^{-\frac{1}{3}} - x^{-\frac{2}{3}}\big), & 1 \leq x \leq 27, \\ 0, & \text{otherwise.} \end{cases}$$

(a) Find the cumulative distribution function of $X$.

The random variable $Y$ is defined by $Y = X^{\frac{1}{3}}$.

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

(c) Find the exact value of the median of $Y$.