11 The continuous random variable \(X\) has probability density function given by
\(f ( x ) = \begin{cases} a x ^ { 2 } & 0 \leqslant x < 2 ,
b ( 3 - x ) ^ { 2 } & 2 \leqslant x \leqslant 3 ,
0 & \text { otherwise } \end{cases}\)
where \(a\) and \(b\) are positive constants.
- Given that \(\mathrm { E } ( X ) = 2\), determine the values of \(a\) and \(b\).
- Determine the median value of \(X\).
- A random sample of 50 observations of \(X\) is selected.
Given that \(\operatorname { Var } ( X ) = 0.2\), determine an estimate of the probability that the mean value of the 50 observations is less than 1.9.