Multiple independent time periods

7 A random variable \(X\) has the distribution \(\operatorname { Po } ( 2.4 )\).

1. Find \(\mathrm { P } ( 2 \leqslant X < 4 )\).
2. Two independent values of \(X\) are chosen. Find the probability that both of these values are greater than 1 .
3. It is given that \(\mathrm { P } ( X = r ) < \mathrm { P } ( X = r + 1 )\).
   1. Find the set of possible values of \(r\).
   2. Hence find the value of \(r\) for which \(\mathrm { P } ( X = r )\) is greatest.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Customers arrive at a shop at a constant average rate of 2.3 per minute.

1. State another condition for the number of customers arriving per minute to have a Poisson distribution.
   It is now given that the number of customers arriving per minute has the distribution \(\mathrm { Po } ( 2.3 )\).
2. Find the probability that exactly 3 customers arrive during a 1 -minute period.
3. Find the probability that more than 3 customers arrive during a 2 -minute period.
4. Five 1-minute periods are chosen at random. Find the probability that no customers arrive during exactly 2 of these 5 periods.

3. Accidents occur randomly at a road junction at a rate of 18 every year. The random variable \(X\) represents the number of accidents at this road junction in the next 6 months.

1. Write down the distribution of \(X\).
2. Find \(\mathrm { P } ( X > 7 )\).
3. Show that the probability of at least one accident in a randomly selected month is 0.777 (correct to 3 decimal places).
4. Find the probability that there is at least one accident in exactly 4 of the next 6 months.

7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,

1. find the probability that the chosen patch contains
   1. no daisies,
   2. one daisy. Ten such patches are chosen. Using your answers to part (a),
2. find the probability that the total number of daisies is less than two.
3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
4. Compare your answers to parts (b) and (c).
5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.

The number of goals scored per match by Everly Rovers is represented by the random variable \(X\) which has mean 1.8.

1. State two conditions for \(X\) to be modelled by a Poisson distribution. [2]

Assume now that \(X \sim \text{Po}(1.8)\).

1. Find \(\text{P}(2 < X < 6)\). [2]
2. The manager promises the team a bonus if they score at least 1 goal in each of the next 10 matches. Find the probability that they win the bonus. [3]

On a stretch of motorway accidents occur at a rate of 0.9 per month.

1. Show that the probability of no accidents in the next month is 0.407, to 3 significant figures. [1] Find the probability of
2. exactly 2 accidents in the next 6 month period, [3]
3. no accidents in exactly 2 of the next 4 months. [3]