Problem

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

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

where $k$ is a constant.

(a) Show that $k=\frac{2}{5}$.

(b) Find the interquartile range of $X$.

(c) Find $\operatorname{Var}(X)$.