7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f3fdcd3c-c1c8-4205-a730-eb0bab8607d4-11_471_816_233_548}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). The part of the sketch from \(x = 0\) to \(x = 4\) consists of an isosceles triangle with maximum at ( \(2,0.5\) ).
- Write down \(\mathrm { E } ( X )\).
The probability density function \(\mathrm { f } ( x )\) can be written in the following form.
$$f ( x ) = \begin{cases} a x & 0 \leqslant x < 2
b - a x & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ - Find the values of the constants \(a\) and \(b\).
- Show that \(\sigma\), the standard deviation of \(X\), is 0.816 to 3 decimal places.
- Find the lower quartile of \(X\).
- State, giving a reason, whether \(\mathrm { P } ( 2 - \sigma < X < 2 + \sigma )\) is more or less than 0.5