(a)[2]
The graph of the function $f$ is a line segment running from $(0, 0)$ to $(2, 1)$. Show that $f$ could be a probability density function.
(b)[2]
The graph of the function $g$ is a semicircle, centre $(0, 0)$, and it lies entirely above the $x$-axis. Given that $g$ is a probability density function, find the radius of the semicircle.
(c(i))[1]
The time, $X$ minutes, taken by a large number of students to complete a test has probability density function $h$, as shown in the diagram. Without calculation, use the diagram to explain how you can tell that the median time is less than $15$ minutes.
(c(ii))[3]
It is now given that $h(x)=\begin{cases} \frac{40}{x^2}-\frac{1}{10} & 10 \le x \le 20, \\ 0 & \text{otherwise}. \end{cases}$ Find the mean time.