9 A rectangle of area \(A \mathrm {~m} ^ { 2 }\) has a perimeter of 20 m and each of the two shorter sides are of length \(X \mathrm {~m}\), where \(X\) is uniformly distributed between 0 and 2 .
- Write down an expression for \(A\) in terms of \(X\), and hence show that \(A = 25 - ( X - 5 ) ^ { 2 }\).
- Write down the probability density function of \(X\).
- Show that the cumulative distribution function of \(A\) is
$$\mathrm { F } ( a ) = \left\{ \begin{array} { l r }
0 & a < 0 ,
\frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 ,
1 & a > 16 .
\end{array} \right.$$ - Find the probability density function of \(A\).
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