\includegraphics[max width=\textwidth, alt={}, center]{3f1a0c67-03a4-4b4f-99c0-4336ba7d56b0-3_255_643_264_790}
The diagram shows the graph of the probability density function, f , of a random variable \(X\), where
$$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 0 & \text { otherwise } \end{cases}$$
State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).
A random variable, \(W\), has probability density function given by
$$\mathrm { g } ( w ) = \begin{cases} a w & 0 \leqslant w \leqslant b 0 & \text { otherwise } \end{cases}$$
where \(a\) and \(b\) are constants. Given that the median of \(W\) is 2 , find \(a\) and \(b\).