Problem

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

$$f(x) = \begin{cases} a, & 0 \leq x \leq 5, \\ b - cx, & 5 \leq x \leq 8, \\ 0, & \text{otherwise,} \end{cases}$$

where $a$, $b$ and $c$ are constants.

(a) Show that $a = \frac{2}{13}$ and find the values of $b$ and $c$.

(b) Find the mean of $X$.

(c) Find the median of $X$.

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

(d) Find the cumulative distribution function for $Y$.