\begin{enumerate}
  \item The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable $X$, with probability density function
\end{enumerate}

$$f ( x ) = \begin{cases} \frac { 3 } { 64 } x ^ { 2 } ( 4 - x ) & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

Using this model, find, by algebraic integration,\\
(a) the mean number of hours that a component will last,\\
(b) the standard deviation of $X$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{ce1f9aa7-cf16-4293-98b1-157eed35b761-06_478_974_1069_479}
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\caption{Figure 1}
\end{center}
\end{figure}

Figure 1 shows a sketch of the probability density function of the random variable $X$.\\
(c) Give a reason why the random variable $X$ might be unsuitable as a model for the time, in thousands of hours, that these electrical components will last.\\
(d) Sketch a probability density function of a more realistic model.

\hfill \mbox{\textit{Edexcel S2 2017 Q4 [10]}}