3. Figure 1 shows an accurate graph of the cumulative distribution function, \(\mathrm { F } ( x )\), for the continuous random variable \(X\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17296edc-9ab4-4f81-ae68-c76190986fd1-08_535_1152_354_342}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
- Find \(\mathrm { P } ( 3 < X < 7 )\)
The probability density function of \(X\) is given by
$$\mathrm { f } ( x ) = \begin{cases} a & 2 \leqslant x < 4
b & 4 \leqslant x < 6
c & 6 \leqslant x \leqslant 8
0 & \text { otherwise } \end{cases}$$
where \(a\), \(b\) and \(c\) are constants. - Find the value of \(a\), the value of \(b\) and the value of \(c\)
- Find \(\mathrm { E } ( X )\)