Problem

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

$$f(x) = \begin{cases} \dfrac{1}{21}(x - 1)^2, & 2 \leq x \leq 5, \\ 0, & \text{otherwise.} \end{cases}$$

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

The random variable $Y$ is defined by $Y = (X - 1)^4$.

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

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

(d) Find $\mathrm{E}(Y)$.