Problem

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

$$f(x) = \begin{cases} kx(4 - x), & 0 \leq x < 2,\\ k(6 - x), & 2 \leq x \leq 6,\\ 0, & \text{otherwise,} \end{cases}$$

where $k$ is a constant.

(a) Show that $k = \frac{3}{40}$.

(b) Given that $\mathrm{E}(X) = 2.5$, find $\operatorname{Var}(X)$.

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