Poisson Distribution

A Poisson experiment is a statistical experiment that has the following properties:

• The experiment results in outcomes that can be classified as successes or failures.
• The average number of successes (μ) that occurs in a specified region is known.
• The probability that a success will occur is proportional to the size of the region.
• The probability that a success will occur in an extremely small region is virtually zero.

Note that the specified region could take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.

Notation

The following notation is helpful, when we talk about the Poisson distribution.

• e: A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.)
• μ: The mean number of successes that occur in a specified region.
• x: The actual number of successes that occur in a specified region.
• P(x; μ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is μ.

Poisson Distribution

A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution.

Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula:

Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:

P(x; μ) = (e^-μ) (μ^x) / x!

where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.

The Poisson distribution has the following properties:

• The mean of the distribution is equal to μ .
• The variance is also equal to μ .

Example 1

The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?

Solution: This is a Poisson experiment in which we know the following:

• μ = 2; since 2 homes are sold per day, on average.
• x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.
• e = 2.71828; since e is a constant equal to approximately 2.71828.

We plug these values into the Poisson formula as follows:

P(x; μ) = (e^-μ) (μ^x) / x!
P(3; 2) = (2.71828^-2) (2³) / 3!
P(3; 2) = (0.13534) (8) / 6
P(3; 2) = 0.180

Thus, the probability of selling 3 homes tomorrow is 0.180 .

Cumulative Poisson Probability

A cumulative Poisson probability refers to the probability that the Poisson random variable is greater than some specified lower limit and less than some specified upper limit.

Example 1

Suppose the average number of lions seen on a 1-day safari is 5. What is the probability that tourists will see fewer than four lions on the next 1-day safari?

Solution: This is a Poisson experiment in which we know the following:

• μ = 5; since 5 lions are seen per safari, on average.
• x = 0, 1, 2, or 3; since we want to find the likelihood that tourists will see fewer than 4 lions; that is, we want the probability that they will see 0, 1, 2, or 3 lions.
• e = 2.71828; since e is a constant equal to approximately 2.71828.

To solve this problem, we need to find the probability that tourists will see 0, 1, 2, or 3 lions. Thus, we need to calculate the sum of four probabilities: P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5). To compute this sum, we use the Poisson formula:

P(x < 3, 5) = P(0; 5) + P(1; 5) + P(2; 5) + P(3; 5)
P(x < 3, 5) = [ (e^-5)(5⁰) / 0! ] + [ (e^-5)(5¹) / 1! ] + [ (e^-5)(5²) / 2! ] + [ (e^-5)(5³) / 3! ]
P(x < 3, 5) = [ (0.006738)(1) / 1 ] + [ (0.006738)(5) / 1 ] + [ (0.006738)(25) / 2 ] + [ (0.006738)(125) / 6 ]
P(x < 3, 5) = [ 0.0067 ] + [ 0.03369 ] + [ 0.084224 ] + [ 0.140375 ]
P(x < 3, 5) = 0.2650

Thus, the probability of seeing at no more than 3 lions is 0.2650.