Deriving CDF from area proportionality

A question is this type if and only if it asks to derive or explain the cumulative distribution function F(r) for a random variable representing distance from center, based on the principle that probability is proportional to area.

2 questions

OCR MEI S3 2012 January Q4
4 At the school summer fair, one of the games involves throwing darts at a circular dartboard of radius \(a\) lying on the ground some distance away. Only darts that land on the board are counted. The distance from the centre of the board to the point where a dart lands is modelled by the random variable \(R\). It is assumed that the probability that a dart lands inside a circle of radius \(r\) is proportional to the area of the circle.
  1. By considering \(\mathrm { P } ( R < r )\) show that \(\mathrm { F } ( r )\), the cumulative distribution function of \(R\), is given by $$\mathrm { F } ( r ) = \begin{cases} 0 & r < 0 ,
    \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leqslant r \leqslant a ,
    1 & r > a . \end{cases}$$
  2. Find \(\mathrm { f } ( r )\), the probability density function of \(R\).
  3. Find \(\mathrm { E } ( R )\) and show that \(\operatorname { Var } ( R ) = \frac { a ^ { 2 } } { 18 }\). The radius \(a\) of the dartboard is 22.5 cm .
  4. Let \(\bar { R }\) denote the mean distance from the centre of the board of a random sample of 100 darts. Write down an approximation to the distribution of \(\bar { R }\).
  5. A random sample of 100 darts is found to give a mean distance of 13.87 cm . Does this cast any doubt on the modelling?
Edexcel S2 Q6
  1. 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 \(R\),
    1. explain why the cumulative distribution function \(\mathrm { F } ( r )\) is given by
    $$\begin{array} { l l } \mathrm { F } ( r ) = 0 & r < 0 ,
    \mathrm {~F} ( r ) = \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leq r \leq a ,
    \mathrm {~F} ( r ) = 1 & r > a . \end{array}$$
  2. 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 $$\mathrm { F } ( x ) = 0 , x < 0 ; \quad \mathrm { F } ( x ) = \frac { x } { a } \left( 2 - \frac { x } { a } \right) , 0 \leq x \leq a ; \quad \mathrm { F } ( x ) = 1 , x > a .$$
  3. Find the probability density function of \(X\), and explain why it shows that Bob is aiming for \(O\).