1. The continuous random variable \(X\) has the following cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1
\frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2
k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4
1 & x > 4 \end{array} \right.$$ where \(k , a\) and \(b\) are constants.
Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)

1. show that \(a = 4\)
2. Find \(\mathrm { P } ( X < 2.5 )\) giving your answer to 3 significant figures.