- The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable \(X\), with probability density function
$$f ( x ) = \begin{cases} \frac { 3 } { 64 } x ^ { 2 } ( 4 - x ) & 0 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
Using this model, find, by algebraic integration,
- the mean number of hours that a component will last,
- the standard deviation of \(X\).
\begin{figure}[h]
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\caption{Figure 1}
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Figure 1 shows a sketch of the probability density function of the random variable \(X\). - 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.
- Sketch a probability density function of a more realistic model.