7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f3fdcd3c-c1c8-4205-a730-eb0bab8607d4-11_471_816_233_548}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\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 ( $2,0.5$ ).
\begin{enumerate}[label=(\alph*)]
\item Write down $\mathrm { E } ( X )$.

The probability density function $\mathrm { f } ( x )$ can be written in the following form.

$$f ( x ) = \begin{cases} a x & 0 \leqslant x < 2 \\ b - a x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
\item Find the values of the constants $a$ and $b$.
\item Show that $\sigma$, the standard deviation of $X$, is 0.816 to 3 decimal places.
\item Find the lower quartile of $X$.
\item State, giving a reason, whether $\mathrm { P } ( 2 - \sigma < X < 2 + \sigma )$ is more or less than 0.5
\end{enumerate}

\hfill \mbox{\textit{Edexcel S2 2009 Q7 [15]}}