- The continuous random variable \(Y\) has probability density function
$$f ( y ) = \left\{ \begin{array} { c c }
\frac { 1 } { 24 } ( y + 2 ) ( 4 - y ) & 0 \leqslant y \leqslant 3
0 & \text { otherwise }
\end{array} \right.$$
- Show that the mode of \(Y\) is 1 , justifying your reasoning.
Given that \(\mathrm { P } ( Y < 1 ) = \frac { 13 } { 36 }\)
- determine whether the median of \(Y\) is less than, equal to, or greater than 2 Give a reason for your answer.
Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 213 } { 80 }\)
- find, using algebraic integration, \(\operatorname { Var } ( 2 Y )\)