Problem

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

$$f(x)= \begin{cases} kx^3, & 0 \le x < 1,\\ k(5-x), & 1 \le x \le 5,\\ 0, & \text{otherwise}, \end{cases}$$

where $k$ is a constant.

(a) Sketch the graph of $f$.

(b) Show that $k=\frac{4}{33}$.

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

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