- The random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by
$$\mathrm { f } ( y ) = \left\{ \begin{array} { c c }
k y ( a - y ) & 0 \leqslant y \leqslant 3
0 & \text { otherwise }
\end{array} \right.$$
where \(k\) and \(a\) are positive constants.
- Explain why \(a \geqslant 3\)
- Show that \(k = \frac { 2 } { 9 ( a - 2 ) }\)
Given that \(\mathrm { E } ( Y ) = 1.75\)
- show that \(a = 4\) and write down the value of \(k\).
For these values of \(a\) and \(k\),
- sketch the probability density function,
- write down the mode of \(Y\).