Problem

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

$$f(x) = \begin{cases} \dfrac{x}{4}(4 - x^2), & 0 \leq x \leq 2, \\ 0, & \text{otherwise.} \end{cases}$$

(a) Find $\operatorname{Var}(\sqrt{X})$.

The continuous random variable $Y$ is defined by $Y = X^2$.

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

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