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Wize University Statistics Textbook > Discrete Probability Distributions

Poisson Distribution

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Poisson Experiment


A Poisson experiment has the following properties:
  1. Each trial has exactly two possible outcomes -- "success" or "failure".
  2. The average number of successes is known and is constant (λ\purple\lambda) from interval to interval of time or space
  • i.e. the number of occurrences from one "block" to another "block" is independent but have the same constant average
  1. The number of occurrences must be uniformly distributed over the entire interval
  • i.e. the probability of success in a small interval is proportional to the size of the entire interval of time or space
  • i.e. The probability that a success will occur in an extremely small region is virtually zero

Watch Out!
"Success" just denotes the outcome that you are interested in.
"Failure" just denotes the opposite outcome (i.e. complement).

For example, these could be "male or female", "win or lose", "greater than 1 or not greater than 1", etc.

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Examples
Determine if the following are Poisson experiments:

1. A certain realtor sells on average 3 houses per month, and you want to find the probability that he sells 4 houses next month →
YES

Let "success" = sold a home and "failure" = didn't sell a home.
The average number of successes in a given interval (month) is 3 (it is known and is a constant).
The probability that a success will occur (the realtor sells a house) is proportional to the interval of time (i.e. the longer the interval, the higher the probability of success)

2. A certain realtor sells on average 1 house during the 1st week of the month, 2 houses during the 2nd week of the month, 3 houses during the 3rd week of the month, and 4 houses during the 4th week of the month -- so on average, he sells 1+2+3+44=2.5\frac{1+2+3+4}{4}=2.5 houses per week. You want to find the probability that she sells 3 houses in the next week →
NO

Depending on which week of the month it is, the probability of success varies -- the average rate of success is not constant and the probability of success is not directly proportional to the size of the time interval (number of weeks). Therefore, this is NOT a Poisson experiment.
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Poisson Distribution

Let XX be a random variable that represents the number of successes in a Poisson experiment where
  • the average rate of success on a ceratin interval is λ\lambda
Then XX follows a Poisson distribution (or Poisson model), and we denote this by X∼Poisson(λ)\boxed{X\sim \text{Poisson}(\lambda)} or X∼P(λ)\boxed{X\sim P(\lambda)}.

The probability of obtaining exactly xx number of successes in an interval is given by this formula:
P(X=x)=λxe−λx!\displaystyle \boxed{P\left(X=x\right)=\frac{\lambda^xe^{-\lambda}}{x!}}

Note: ee is an irrational number,ee ≈ 2.718282

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Mean, Variance, and Standard Deviation

  • Mean: E(X)=λ\boxed{E(X)=\lambda}
  • Variance: Var(X)=σ2=λ\boxed{Var(X)=\sigma^2=\lambda}
  • Standard deviation: SD(X)=σ=λ\boxed{SD(X)=\sigma=\sqrt {\lambda}}
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Example: Poisson Probability

A doctor's office sees 8 patients every hour between 9am and 10am every day. Find the probability that this office sees between 5 and 7 patients tomorrow between 9am and 10am.

Let "success" = a patient is seen by the doctor and "failure" = a patient is not seen by the doctor.

The average number of patients in this one hour interval is constant -- λ=8\lambda=8.

The probability of success is proportional to the size of the interval (i.e. the probability of a patient being seen by the doctor increases proportionately with the number of day you perform this experiment)

So, this is a Poisson experiment.

Using our formula:
P(5≤X≤7)P\left(5\le X\le7\right)

=P(X=5)+P(X=6)+P(X=7)=P\left(X=5\right)+P\left(X=6\right)+P\left(X=7\right)

=85e−85!+86e−86!+87e−87!\displaystyle =\frac{8^5e^{-8}}{5!}+\frac{8^6e^{-8}}{6!}+\frac{8^7e^{-8}}{7!}

≈0.353\approx 0.353

Therefore, the probability of getting between 5 and 7 patients tomorrow between 9am and 10am is approximately 0.353.

Andy is a rude bank teller. He gets an average of 3 complaints per 8-hour shift.

a) Determine the mean and standard deviation for the number of complaints in one hour.

b) What is the probability that he will get one complaint in the first hour?

c) What is the probability of Andy receiving no complaints during an 8-hour shift?

The average number of homes sold by Jon Jennings, a top realtor, is 14 homes per week. Assume he works all 7 days and all days are the same.

a) What is the probability that exactly 3 homes will be sold tomorrow?

b) What is the probability that at least 3 homes will be sold tomorrow?

c) Jon wishes to take 2 days off next week but he still wants to sell 14 homes. What is the likelihood that he can achieve this?
Extra Practice
Poisson Distribution
The average number of homes sold by Jon Jennings, a top realtor, is 14 homes per week. Assume he works all 7 days and all days are the same.
a) What is the probability that exactly 3 homes will be sold tomorrow?
b) What is the probability that at least 3 homes will be sold tomorrow?
Poisson Model
Andy is a rude bank teller. He gets an average of 3 complaints per 8-hour shift.
a) Determine the mean and standard deviation for the number of complaints in one hour.
b) What is the probability that he will get one complaint in the first hour?
Poisson Distribution
An injury is reported in Company A once a week. In Company B, an injury is reported only once a month. All injuries are reported and the occurrences of injuries in those companies are independent. In the next three weeks, what is the probability that Company A reports fewer than 2 injuries and Company B reports no injuries?
Let 1 month = 4 weeks
Binomial Distribution vs. Poisson Distribution vs. Hypergeometric Probability
Binomial [B], Poisson [P], or Hypergeometric [H]? Enter B, P, or H in the boxes below.
Binomial and Poisson Distributions
At a dangerous intersection, 2.5% of cars drive through a red light. Suppose 100 cars drove by.
Poisson Distribution
Which of the following can be modelled with a Poisson distribution?
Poisson Distribution
A random variable X has a poisson distribution X ∼Poisson(2)X\ \sim Poisson\left(2\right). Find the probability that XX is at least 1.
Poisson distribution
An old apartment building experiences an average of 2 electrical failure a week. Determine the probability that there will be no electrical failure on a given day.
Poisson Distribution
The number of customers coming to Suzanne’s coffee shop has a mean of 6 customers per hour. What is the probability that at least one customer will come to her coffee shop in an interval of 30 minutes?
Poisson Distribution
Which of the following cannot be modelled with a Poisson distribution?
Poisson Distribution
Five paper cuts are reported to the HR manager each 7-day week, on average. What is the probability that at least one paper cut will be reported today?
Poisson distribution
The Moody Volcano in Hawaii erupts four times a year. What is the probability that it will not erupt in the next month?
Statistical Properties: Practice
True or false? Enter "T" for true; "F" for false (without quotations).
Poisson Distribution
Five injuries are reported at the gym each day. What is the probability that fewer than 3 injuries occur today?