5 The continuous random variable \(X\) has cumulative distribution function given by
$$F ( x ) = \begin{cases} 0 & x < 1 ,
\frac { 1 } { 8 } ( x - 1 ) ^ { 2 } & 1 \leqslant x < 3 ,
a ( x - 2 ) & 3 \leqslant x < 4 ,
1 & x \geqslant 4 , \end{cases}$$
where \(a\) is a positive constant.
- Find the value of \(a\).
- Verify that \(C _ { X } ( 8 )\), the 8th percentile of \(X\), is 1.8 .
- Find the cumulative distribution function of \(Y\), where \(Y = \sqrt { X - 1 }\).
- Find \(C _ { Y } ( 8 )\) and verify that \(C _ { Y } ( 8 ) = \sqrt { C _ { X } ( 8 ) - 1 }\).