Problem

The continuous random variable $X$ has probability density function $\mathrm{f}$ given by

$$\mathrm{f}(x)= \begin{cases} \frac{4}{9}(x+1), & 0\leq x\leq 1,\\ (x-2)^2, & 1<x\leq 2,\\ 0, & \text{otherwise.} \end{cases}$$

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

(b) Find the exact value of the median of $X$.

The random variable $Y$ is defined by $Y=\sqrt{X}$.

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

(d) Determine whether the median of $Y$ is greater than, or less than, the median of $X$.