Problem

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

$$f(x) = \begin{cases} \dfrac{3}{8}\bigg(1 + \dfrac{1}{x^2}\bigg), & 1 \leq x \leq 3,\\ 0, & \text{otherwise.} \end{cases}$$

(a) Find $\mathrm{E}(\sqrt{X})$.

The random variable $Y$ is given by $Y = X^2$.

(b) Find the probability density function of $Y$.

(c) Find the 40th percentile of $Y$.