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\includegraphics[max width=\textwidth, alt={}, center]{332f0909-c192-40f7-88b7-7cfec2db2eef-06_428_773_260_685}
The time, \(X\) minutes, taken by a large number of runners to complete a certain race has probability density function f given by
$$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10
0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant, as shown in the diagram.
- Without calculation, explain how you can tell that there were more runners whose times were below 7.5 minutes than above 7.5 minutes.
- Show that \(k = 10\).
- Find \(\mathrm { E } ( X )\).
- Find \(\operatorname { Var } ( X )\).