3 Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.
| \(\boldsymbol { x }\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\) | 0 | 0.19 | 0.26 | 0.20 | 0.13 | 0.07 | 0.15 |
- Find the probability that a member borrows more than 3 books.
- Assume that the numbers of books borrowed by two particular members are independent.
Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books.
- Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\).
- One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\).
Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\).
- The library introduces a fee of 10 pence for each book borrowed.
Assuming that the probabilities do not change, calculate:
- the mean amount that will be paid by a member;
- the standard deviation of the amount that will be paid by a member.