Mixed practice: Poisson distribution Problem set A

X ~ Po(6.5).

1. (a) Find P(X = 5).
2. (b) Find P(X ≤ 8).

X ~ Po(3.7).

1. (a) Find P(X < 4).
2. (b) Find P(X ≥ 2).
3. (c) Find P(X > 5).

X ~ Po(9.2).

1. (a) Find P(6 ≤ X ≤ 10).
2. (b) Find P(6 < X < 10).
3. (c) Find P(6 ≤ X < 10).

X ~ Po(7.7). Write down

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

A help desk receives telephone calls at a mean rate of 21 per hour. The number of calls in any period has a Poisson distribution.

1. (a) State the distribution of the number of calls in a 20 minute period.
2. (b) Find the probability of exactly 9 calls in a 20 minute period.
3. (c) Find the probability of no calls in a 5 minute period.

Customers enter a shop through the front door at a mean rate of 7.3 per hour and through the side door at a mean rate of 4.1 per hour. Each number of arrivals 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 more than 14.
3. (c) Give one reason why the two counts may not be independent.

The number of support tickets that a website receives in one day has a Poisson distribution with mean 9.4.

1. (a) Find the probability that there are more than 12 tickets in one day.
2. (b) Find the probability that there are more than 12 tickets on each of three consecutive days.

The number of faults on a machine in one day has a Poisson distribution. The probability of at least one fault in a day is 0.7.

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

The number of landslides in a valley in one year has a Poisson distribution. The probability of more than 2 landslides in a year is 0.35.

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

The number of ink smudges on a printed page has a Poisson distribution with mean 2.6.

1. (a) Find the most likely number of smudges on a page. Justify your answer.
2. (b) Find the probability that there are at least 2 smudges on a page.
3. (c) There are 14 pages. Find the probability that fewer than half of them have at least 2 smudges on a page.

The number of customers arriving at a kiosk in a 5 minute period has a Poisson distribution with mean 2.7.

1. (a) Find the probability that at least 2 arrive.
2. (b) Given that at least 2 arrive, find the probability that exactly 4 arrive.

A tool shop has 2 pressure washers to hire by the day. The number of requests per day has a Poisson distribution with mean 1.7. On a day with one request, one washer is hired. On a day with 2 or more requests, both are hired and the extra requests are refused. Assume the two washers are hired equally often.

1. (a) Find the probability that no pressure washer is hired.
2. (b) Find the probability that some requests have to be refused.
3. (c) Over 250 days, find the expected number of days on which a particular pressure washer is in use.

A florist sells bouquets at a mean rate of 3.7 per day, with a Poisson distribution. A delivery of bouquets arrives every 4 days and the stock must last until the next delivery.

1. (a) State the distribution of the number of bouquets demanded in 4 days.
2. (b) Find the smallest number of bouquets that should be in stock at the start so that the probability that demand is greater than the stock is at most 0.05.