4 The random variable \(X\) has probability density function given by
$$f ( x ) = \begin{cases} \frac { k } { ( x + 1 ) ^ { 2 } } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$
where \(k\) is a constant.
- Show that \(k = 2\).
- Find \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 1 } { 5 }\).
\includegraphics[max width=\textwidth, alt={}, center]{18cef198-5ca2-4700-88e9-1a2bd55f841e-2_367_524_1548_849}
The diagram shows the graph of \(y = \mathrm { f } ( x )\). The median of \(X\) is denoted by \(m\). Use the diagram to explain whether \(m < 0.5\), \(m = 0.5\) or \(m > 0.5\).