1. A continuous random variable \(Y\) has cumulative distribution function given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { c r } 0 & y < 3
\frac { 1 } { 16 } \left( y ^ { 2 } - 6 y + a \right) & 3 \leqslant y \leqslant 5
\frac { 1 } { 12 } ( y + b ) & 5 < y \leqslant 9
\frac { 1 } { 12 } \left( 100 y - 5 y ^ { 2 } + c \right) & 9 < y \leqslant 10
1 & y > 10 \end{array} \right.$$ where \(a\), \(b\) and \(c\) are constants.

1. Find the value of \(a\) and the value of \(c\)
2. Find the value of \(b\)
3. Find \(\mathrm { P } ( 6 < Y \leqslant 9 )\) Show your working clearly.
4. Specify the probability density function, f(y), for \(5 < y \leqslant 9\) Using the information $$\int _ { 3 } ^ { 5 } ( 6 y - 5 ) f ( y ) d y + \int _ { 9 } ^ { 10 } ( 6 y - 5 ) f ( y ) d y = 26.5$$
5. find \(\mathrm { E } ( 6 Y - 5 )\) You should make your method clear.