Week 1 - Lesson 4.6 Poisson Probability Distribution Flashcards
This distribution is useful for describing the number of events that will occur during a specific
interval of time or in a specific distance, area, or volume.
Poisson probability distribution
A Poisson probability distribution is useful for describing the number of events that will occur during a specific
interval of time or in a specific distance, area, or volume. Examples of such random variables are:
The number of traffic accidents at a particular intersection
The number of house fire claims per month that are received by an insurance company
The number of people who are infected with the AIDS virus in a certain neighborhood
The number of people who walk into a barber shop without an appointment
In a binomial distribution, if the number of trials, n, gets larger and larger as the probability of success, p, gets
smaller and smaller, we obtain a Poisson distribution. The section below lists some of the basic characteristics of a
Poisson distribution.
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What are the Characteristics of a Poisson Distribution
• The experiment consists of counting the number of events that will occur during a specific interval of time or
in a specific distance, area, or volume.
• The probability that an event occurs in a given time, distance, area, or volume is the same.
• Each event is independent of all other events. For example, the number of people who arrive in the first hour
is independent of the number who arrive in any other hour.
The probability distribution, mean, and variance of a Poisson random variable are given as follows:
formula on module
Example: A lake, popular among boat fishermen, has an average catch of three fish every two hours during the month
of October.
a. What is the probability distribution for X, the number of fish that you will catch in 7 hours?
b. What is the probability that you will catch 0 fish in seven hours of fishing? What is the probability of catching 3
fish? How about 10 fish?
c. What is the probability that you will catch 4 or more fish in 7 hours?
a. The mean number of fish is 3 fish in 2 hours, or 1.5 fish/hour. This means that over seven hours, the mean number
of fish will be l = 1.5 fish/hour • 7 hours = 10.5 fish. Thus, the equation becomes:
b. To calculate the probabilities that you will catch 0, 3, or 10 fish, perform the following calculations
c. The probability that you will catch 4 or more fish in 7 hours is equal to the sum of the probabilities that you will
catch 4 fish, 5 fish, 6 fish, and so on, as is shown below:
The Complement Rule can be used to find this probability as follows:
Therefore, there is about a 99% chance that you will catch 4 or more fish within a 7 hour period during the month of
October.
Example: A zoologist is studying the number of times a rare kind of bird has been sighted. The random variable X
is the number of times the bird is sighted every month. We assume that X has a Poisson distribution with a mean
value of 2.5.
a. Find the mean and standard deviation of X.
b. Find the probability that exactly five birds are sighted in one month.
c. Find the probability that two or more birds are sighted in a 1-month period.
a. The mean and the variance are both equal to l. Thus, the following is true:
b. Now we want to calculate the probability that exactly five birds are sighted in one month. For this, we use the
Poisson distribution formula:
c. The probability of two or more sightings is an infinite sum and is impossible to compute directly. However, we
can use the Complement Rule as follows:
Therefore, according to the Poisson model, the probability that two or more sightings are made in a month is 0.713.
Technology Note: Calculating Poisson Probabilities on the TI-83/84 Calculator
Press [2ND][DIST] and scroll down to ’poissonpdf(’. Press [ENTER] to place ’poissonpdf(’ on your home screen.
Type values of µ and x, separated by commas, and press [ENTER].
Use ’poissoncdf(’ for the probability of at most x successes.
Note: It is not necessary to close the parentheses.
Technology Note: Using Excel
In a cell, enter the function =Poisson(µ, x,false), where µ and x are numbers. Press [ENTER], and the probability of
x successes will appear in the cell.
For the probability of at least x successes, replace ’false’ with ’true’.
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Characteristics of a Poisson distribution:
• The experiment consists of counting the number of events that will occur during a specific interval of time or
in a specific distance, area, or volume.
• The probability that an event occurs in a given time, distance, area, or volume is the same.
• Each event is independent of all other events.
What is Poisson random variable formula
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