3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e4af10e-ee8d-493f-bd72-34b231003d97-05_455_1026_242_484}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\).
For \(0 \leqslant x \leqslant 3 , \mathrm { f } ( x )\) is represented by a curve \(O B\) with equation \(\mathrm { f } ( x ) = k x ^ { 2 }\), where \(k\) is a constant.
For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm { f } ( x )\) is represented by a straight line passing through \(B\) and the point ( \(a , 0\) ).
For all other values of \(x , \mathrm { f } ( x ) = 0\).
Given that the mode of \(X =\) the median of \(X\), find
- the mode,
- the value of \(k\),
- the value of \(a\).
Without calculating \(\mathrm { E } ( X )\) and with reference to the skewness of the distribution
- state, giving your reason, whether \(\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3\) or \(\mathrm { E } ( X ) > 3\).