3. The random variable \(X\) has probability density function given by
$$f ( x ) = \begin{cases} a x + b & 1 \leqslant x < 4
\frac { 3 } { 2 } - \frac { 1 } { 4 } x & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$
as shown in Figure 1, where \(a\) and \(b\) are constants.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a1534ea-4c62-4945-850a-9460ea87fd64-08_634_1132_694_397}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
- Show that the median of \(X\) is 4
- Find the value of \(a\) and the value of \(b\)
- Specify fully the cumulative distribution function of \(X\)