**POISSON**

The Poisson distribution applies when:

NOTATION

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

A constant equal to approximately 2.71828.

The mean number of successes that occur in a specified region.

The actual number of successes that occur in a specified region.

P(x; λ): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is λ.

Poisson Distribution formula

**EXAMPLE 1**

Poisson

A Poisson distribution is the probability distribution that results from a Poisson experiment.

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 ( λ) 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.

1. The event is something that can be counted in whole numbers.

2. Occurrences are independent, so that one occurrence neither diminishes nor increases the chance of another.

3. The average frequency of occurrence for the time period in question is known.

4. Is possible to count how many events have occurred.

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

The Poisson distribution has the following properties:

The mean of the distribution is equal to λ.

The variance is also equal to λ.

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?

λ = 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; λ) = (e- λ) ( λx) / 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 .

HISTORY

The Poisson Distribution is named for its discoverer, who first applied it to the deliberations of juries; in that form it did not attract wide attention. More suggestive was Poisson's application to the science of artillery. The distribution was later and independently discovered by von Bortkiewicz, Rutherford, and Gosset.It was von Bortkiewicz who called it The Law of Small Numbers, but as noted above, though it has a special usefulness at the small end of the range, a Poisson Distribution may also be computed for larger r. The fundamental trait of the Poisson is its asymmetry, and this trait it preserves at any value of r.

Probability and Statistics I

"POISSON DISTRIBUTION"

Daffne Alba 271693

Ilse Porras 262037

Itzamara Gomez 276788

Raul Gonzalez 260062

Obed Gonzalez 271403

Teacher: Olga Olivas

EXAMPLE 2

Births in a hospital occur randomly at an average rate of 1.8 births per hour.

What is the probability of observing 4 births in a given hour at the hospital?

What about the probability of observing more than or equal to 2 births in a given

hour at the hospital?

What is the probability that we observe 5 births in a given 2 hour interval?

EXAMPLE 3

If the probability is 0.005 that a person attending a parade on a very hot summer day suffer insolation. What is the probability that 18 of the 3,000 people attending the parade suffer insolation?

𝑛=3,000

𝑥=18

𝑝=0.005

𝜆=3000(0.005)=15

EXAMPLE 4

There are 50 misprints in a book which has 250 pages. Find the probability that page 100 has no misprints.

The average number of misprints on a page is 50/250 = 0.2 . Therefore, if we let X be the random variable denoting the number of misprints on a page, X will follow a Poisson distribution with parameter 0.2 .

Date: May 21

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