2.1 Poisson Distribution

Blue, a dog, barks an average of 25 times every hour. His owner, Hayley, uses a Poisson distribution to model the number of times that Blue barks.

Write down two assumptions Hayley has made about Blue’s barks in order to use a Poisson distribution.

Find the probability that Blue barks:

Grace, a grumpy toddler, attends nursery five days a week. The number of tantrums that Grace has in a day follows a Poisson distribution with variance 3.14.

Find the probability that Grace has exactly 17 tantrums during a week at nursery.

Find the probability that Grace has fewer than four tantrums in a two-day period at nursery.

Given that Grace has fewer than four tantrums at nursery one day, find the probability that she had no tantrums at nursery that day.

The table below shows the number of detentions per school day that Ms Ottway, a teacher, issues to students over a period of 150 days. 

By calculating unbiased estimates for the mean and the variance, show that a Poisson distribution is an appropriate model for the number of detentions Ms Ottway issues.

Using a Poisson distribution with the unbiased estimate of the mean, find the probability that Ms Ottway issues at least 5 detentions in a day.

Students at Ms Ottway’s school attend school 5 days a week for 40 weeks a year.

Estimate the number of weeks in a school year that Ms Ottway issues fewer than 3 detentions.

are random variables with .

Given that , find the value of .

Given that , find the value of .

Given that , find the value of .

In a game, enemies appear independently and randomly at an average rate of 2.5 enemies every minute.

Find the probability that exactly 10 enemies will appear in a five-minute period.

Find the probability that at least 3 enemies will appear in a 90-second period.

The probability that at least one enemy appears in  minutes is 0.999.  Find the value of .

Matt has three best friends: Danny, Jessica and Luke. Matt receives messages from Danny at an average rate of 1.7 messages per hour, from Jessica at an average rate of 1.5 messages per hour, and from Luke at an average rate of 1.1 messages per hour.

Stating any necessary assumptions, find the probability that Matt receives fewer than 4 messages from Danny in a two-hour period.

Stating any additional assumptions needed, find the probability that Matt receives exactly 6 messages in total within a one-hour period.

Find the probability that Matt will receive at least one message from any of the three friends within a one-hour period.

Find the probability that Matt will receive at least one message from each friend within a one-hour period.

Lucy loves the cinema and goes on average four times a week. The number of times she goes to the cinema in a week can be modelled as a Poisson distribution with a mean of four times.

Find the probability that Lucy goes to the cinema exactly five times in a week.

Find the probability that Lucy goes to the cinema no more than four times in a fortnight.

Find the probability that Lucy goes to the cinema at least once in a day.

Comic Stans is a comic book store in the city of Krakoa. Customers enter the store randomly and independently at an average rate of 8 people every 15 minutes.

Find the probability that exactly three people enter the store in a 1-minute period.

Find the probability that someone enters the store in a 15-second period.

Find the probability that at most three people enter the store in a 10-minute period.

Find the variance of the number of people entering the store in a 1-hour period.

Amber suggests that she can model the number of times that she hiccups using a Poisson distribution.

Write down two conditions that must apply for this model to be applicable.

The mean number of hiccups in a 30-second period is 2.9.

Assuming a Poisson distribution is applicable, find the probability that

The table below shows the data from a sample of 50 observations of a variable .

	0	1	2	3	4
Frequency	3	5	17	15	10

The table below shows the data from a sample of 100 observations of a variable .

	0	1	2	3	4	5
Frequency	15	19	25	21	12	8

Jim is a bird watcher and is trying to model the number of birds that fly past his window. During a 10-minute period he records the number of birds that fly past his window, and he repeats this a total of 120 times to form a sample.

Calculate unbiased estimates for the mean and the variance for the number of birds that fly past Jim’s window in a ten-minute period.

Explain why a Poisson distribution would be appropriate to model the number of birds that fly past Jim’s window in a 10-minute period.

Jim uses the distribution to model the number of birds that fly past his window in a 10-minute period.

Use Jim’s model to calculate the probability that:

Roberto orders a pizza from Pizza Palace and asks for two types of meat toppings: ham and salami. It is known that the number of pieces of ham that Pizza Palace put on their pizzas follows a Poisson distribution with a mean of 6.2 pieces per pizza. It is also known that the number of pieces of salami on their pizzas follows a Poisson distribution with a mean of 4.9 pieces per pizza. The ham and salami are put on the pizza independently.

Write down the distribution that can be used to model the total number of pieces of meat on Roberto’s pizza.

Find the probability that Roberto’s pizza contains a total of exactly 10 pieces of meat.

Find the probability that Roberto’s pizza contains more than 9 but fewer than 13 pieces of meat.

The random variable .

Given that is a non-negative integer, find the largest value of  such that .

Hence find the mode of .

The random variable  where  is a positive integer.

Show that .

The random variables .

Given that , find the value of .

Given that , find the value of .

Given that where  is a positive integer, find an expression for  in terms of .

Roger is investigating historical data from World War II. In south London, an area of 144 km² was divided into 576 equal squares and it was recorded how many times each square was hit by a flying bomb over a period of time. During this time, it was recorded that a total of 537 bombs landed in the area. The results are as shown in the table below:

Roger suggests that a Poisson distribution with mean of would be an appropriate model for the number of times a square was hit by a flying bomb.

The number of mistakes made by a student, Priya, in a 20-minute revision period is modelled as a Poisson distribution with mean of 1.2. The number of mistakes made by a different student, Qays, in a 30-minute revision period is modelled as a Poisson distribution with a mean of 2.2.

Find the probability that Priya makes exactly 2 mistakes and Qays makes exactly 1 mistake within a one-hour revision period. Write your answer in the form  where  are integers to be found. State any assumptions that are needed.

The number of mistakes made by Priya and Qays in a one-hour revision period are added together. Given that they make exactly 9 mistakes in total in a one-hour revision period, find the probability that Priya made exactly 5 mistakes in that same revision period.

Given that Priya makes exactly 5 mistakes in a one-hour revision period, find the probability that Priya and Qays made exactly 9 mistakes in total in that same revision period.

In a week before a test, Priya has ten 30-minute revision periods. Estimate the number of these revision periods during which Priya will make a mistake.

Whilst writing an essay, Gamu notices that she makes spelling mistakes at a rate of 7 for every 150 words. Gamu models the number of spelling mistakes she makes using a Poisson distribution.

Find the maximum number of words Gamu can write before the probability of her making a spelling mistake exceeds 0.75.

Gamu is asked to write three short essays by her lecturer. She writes one containing 100 words, one containing 200 words and one containing 250 words. An essay is returned by Gamu’s lecturer if more than 1% of its words contain spelling mistakes.

Find the probability that all three short essays are returned.

Use the fact that  is a valid probability distribution, along with the properties of the Poisson distribution, to demonstrate the validity of the Taylor series expansion

for all positive values of .