Problem

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

$$f(x)= \begin{cases} kx, & 0 \leq x < 1,\\ kx^2, & 1 \leq x \leq 2,\\ 0, & \text{otherwise}. \end{cases}$$

(a) Show that $k=\frac{6}{17}$.

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

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

(d) Find $E\bigl(\frac{1}{X}\bigr)$.