Problem

As shown in the diagram, the continuous random variable $X$ has probability density function $f$ given by

$$f(x)= \begin{cases} mx, & 0 \leq x \leq 2,\\ \dfrac{k}{x^2}+c, & 2 \leq x \leq 6,\\ 0, & \text{otherwise}, \end{cases}$$

where $m$, $k$ and $c$ are constants.

(a) Given that $\mathrm{P}(X \leq 2)=\frac{1}{3}$, show that $m=\frac{1}{6}$ and find the values of $k$ and $c$.

(b) Find the exact numerical value of the interquartile range of $X$.