- The continuous random variable \(Y\) has the following probability density function
$$f ( y ) = \begin{cases} \frac { 6 } { 25 } ( y - 1 ) & 1 \leqslant y < 2
\frac { 3 } { 50 } \left( 4 y ^ { 2 } - y ^ { 3 } \right) & 2 \leqslant y < 4
0 & \text { otherwise } \end{cases}$$
- Sketch f(y)
- Find the mode of \(Y\)
- Use algebraic integration to calculate \(\mathrm { E } \left( Y ^ { 2 } \right)\)
Given that \(\mathrm { E } ( Y ) = 2.696\)
- find \(\operatorname { Var } ( Y )\)
- Find the value of \(y\) for which \(\mathrm { P } ( Y \geqslant y ) = 0.9\)
Give your answer to 3 significant figures.