Problem

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

$$f(x)= \begin{cases} \frac{1}{16}\sqrt{x}, & 0 \le x < 4,\\ \frac{1}{k\sqrt{x}}, & 4 \le x \le 9,\\ 0, & \text{otherwise}, \end{cases}$$

where $k$ is a constant.

(a) Show that $k=3$.

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

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

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