Mixed practice: Poisson distribution Problem set B

X ~ Po(4.7).

1. (a) Find P(X = 3).
2. (b) Find P(X ≤ 6).

X ~ Po(2.9).

1. (a) Find P(X < 3).
2. (b) Find P(X ≥ 4).
3. (c) Find P(X > 6).

X ~ Po(7.6).

1. (a) Find P(4 ≤ X ≤ 8).
2. (b) Find P(4 < X < 8).
3. (c) Find P(4 ≤ X < 8).

X ~ Po(12.4). Write down

1. (a) E(X).
2. (b) Var(X).
3. (c) the standard deviation of X.

A café's till registers card payments at a mean rate of 27 per hour. The number of payments in any period has a Poisson distribution.

1. (a) State the distribution of the number of payments in a 20 minute period.
2. (b) Find the probability of exactly 11 payments in a 20 minute period.
3. (c) Find the probability of no payments in a 5 minute period.

Calls to a helpline arrive on two lines. Line 1 receives calls at a mean rate of 5.2 per hour and line 2 at a mean rate of 3.8 per hour. Each number of calls has a Poisson distribution and the two are independent.

1. (a) State the distribution of the total number of arrivals, T, in one hour.
2. (b) Find the probability that the total is at least 12.
3. (c) Give one reason why the two counts may not be independent.

The number of parcels delivered to a depot in one day has a Poisson distribution with mean 11.3.

1. (a) Find the probability that there are more than 14 parcels in one day.
2. (b) Find the probability that there are more than 14 parcels on each of three consecutive days.

The number of rockfalls on a mountain road in one month has a Poisson distribution. The probability of at least one rockfall in a month is 0.55.

1. (a) Find the mean λ.
2. (b) Find the probability that there are more than 2 rockfalls in a month.

The number of power cuts in a village in one month has a Poisson distribution. The probability of more than 2 power cuts in a month is 0.15.

1. (a) Find the mean λ.
2. (b) Find the probability that there are fewer than 2 power cuts in a month.
3. (c) Say whether a Poisson distribution is likely to be a good model, giving a reason.

The number of spelling mistakes in a pupil's essay has a Poisson distribution with mean 1.9.

1. (a) Find the most likely number of mistakes in an essay. Justify your answer.
2. (b) Find the probability that there are at least 2 mistakes in an essay.
3. (c) There are 16 pupils. Find the probability that fewer than half of them have at least 2 mistakes in an essay.

The number of seagulls landing on a pier in a 10 minute period has a Poisson distribution with mean 3.4.

1. (a) Find the probability that at least 3 arrive.
2. (b) Given that at least 3 arrive, find the probability that exactly 5 arrive.

A boat hire firm has 2 canoes to hire by the day. The number of requests per day has a Poisson distribution with mean 0.9. On a day with one request, one canoe is hired. On a day with 2 or more requests, both are hired and the extra requests are refused. Assume the two canoes are hired equally often.

1. (a) Find the probability that no canoe is hired.
2. (b) Find the probability that some requests have to be refused.
3. (c) Over 180 days, find the expected number of days on which a particular canoe is in use.

A greengrocer sells trays of strawberries at a mean rate of 2.6 per day, with a Poisson distribution. A delivery arrives every 6 days and the stock must last until the next delivery.

1. (a) State the distribution of the number of trays demanded in 6 days.
2. (b) Find the smallest number of trays that should be in stock at the start so that the probability that demand is greater than the stock is at most 0.05.