Two people are playing darts. Peg hits points randomly on the circular board, whose radius is \(a\). If the distance from the centre \(O\) of the point that she hits is modelled by the variable \(R\),
explain why the cumulative distribution function \(\mathrm { F } ( r )\) is given by
$$\begin{array} { l l }
\mathrm { F } ( r ) = 0 & r < 0 ,
\mathrm {~F} ( r ) = \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leq r \leq a ,
\mathrm {~F} ( r ) = 1 & r > a .
\end{array}$$
By first finding the probability density function of \(R\), show that the mean distance from \(O\) of the points that Peg hits is \(\frac { 2 a } { 3 }\).
Bob, a more experienced player, aims for \(O\), and his points have a distance \(X\) from \(O\) whose cumulative distribution function is
$$\mathrm { F } ( x ) = 0 , x < 0 ; \quad \mathrm { F } ( x ) = \frac { x } { a } \left( 2 - \frac { x } { a } \right) , 0 \leq x \leq a ; \quad \mathrm { F } ( x ) = 1 , x > a .$$
Find the probability density function of \(X\), and explain why it shows that Bob is aiming for \(O\).