- Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below.
| \(n\) | 0 | 1 | 2 | 3 | 4 | 5 |
| \(\mathrm { P } ( N = n )\) | \(a\) | 0.2 | 0.05 | 0.25 | \(b\) | \(c\) |
Given that \(\mathrm { E } ( 4 N + 2 ) = 14.8\) and \(\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }\)
- show that \(\operatorname { Var } ( N ) = 2.76\)
The group decided to charge a 50p entry fee for the first photograph entered and then 20p for each extra photograph entered into the competition up to a maximum of \(\pounds 1\) per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £1
Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
- calculate the expected entry fee per member.
Bai suggests that, as the mean and variance are close, a Poisson distribution could be used to model the number of photographs entered by a member next year.
- State a limitation of the Poisson distribution in this case.