Skip to content Skip to navigation

Connexions

You are here: Home » Content » Discrete Random Variables: Poisson (optional)

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • CCOT Project display tagshide tags

    This module is included in aLens by: CC Open Textbook ProjectAs a part of collection:"Collaborative Statistics"

    Click the "CCOT Project" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection:"Collaborative Statistics"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection:"Collaborative Statistics"

    Comments:

    "Collaborative Statistics was written by two faculty members at De Anza College in Cupertino, California. This book is intended for introductory statistics courses being taken by students at two- […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Lucy Van Pelt display tagshide tags

    This module is included inLens: Lucy's Lens
    By: Tahiya MaromeAs a part of collection:"Collaborative Statistics"

    Comments:

    "Part of the Books featured on Community College Open Textbook Project"

    Click the "Lucy Van Pelt" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Educational Technology Lens display tagshide tags

    This module is included inLens: Educational Technology
    By: Steve WilhiteAs a part of collection:"Collaborative Statistics"

    Click the "Educational Technology Lens" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • SHN CNX Workshop display tagshide tags

    This module is included inLens: Stategic Horizon Network Workshop on Alternative Couseware -- Connexions Session
    By: ConnexionsAs a part of collection:"Collaborative Statistics"

    Comments:

    "This textbook is used in many sections of community college statistics around the country. The PDF version of the book is also available in the Florida Orange Grove repository."

    Click the "SHN CNX Workshop" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

  • Bio 502 at CSUDH display tagshide tags

    This module is included inLens: Bio 502
    By: Terrence McGlynnAs a part of collection:"Collaborative Statistics"

    Comments:

    "This is the course textbook for Biology 502 at CSU Dominguez Hills"

    Click the "Bio 502 at CSUDH" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Discrete Random Variables: Poisson (optional)

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Summary: This module describes the characteristics of a Poisson experiment and the Poisson probability distribution. This module is included in the Elementary Statistics textbook/collection as an optional lesson.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Characteristics of a Poisson experiment are:

  1. You are interested in the number of times something happens in a certain interval. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are 5 words spelled incorrectly in 100 pages. The interval is the 100 pages.
  2. The Poisson may be derived from the binomial if the probability of success is "small" (such as 0.01) and the number of trials is "large" (such as 1000). You will verify the relationship in the homework exercises. nn is the number of trials and pp is the probability of a "success."
The outcomes of a Poisson experiment fit a Poisson probability distribution. The random variable X=X= the number of occurrences in the interval of interest. The mean and variance are given in the summary.

Example 1

The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. What is the probability that the number of loaves put on the shelf in 5 minutes is 3? Of interest is the number of loaves of bread put on the shelf in 5 minutes. The time interval of interest is 5 minutes.

Let XX = the number of loaves of bread put on the shelf in 5 minutes. If the average number of loaves put on the shelf in 30 minutes (half-hour) is 12, then the average number of loaves put on the shelf in 5 minutes is

( 530 )⋅12=2 ( 530 )⋅12 2 loaves of bread

The probability question asks you to find P(X = 3)P(X = 3).

Example 2

A certain bank expects to receive 6 bad checks per day. What is the probability of the bank getting fewer than 5 bad checks on any given day? Of interest is the number of checks the bank receives in 1 day, so the time interval of interest is 1 day. Let XX = the number of bad checks the bank receives in one day. If the bank expects to receive 6 bad checks per day then the average is 6 checks per day. The probability question asks for P(X<5) P(X 5) .

Example 3

Your math instructor expects you to complete 2 pages of written math homework every day. What is the probability that you complete more than 2 pages a day?

This is a Poisson problem because your instructor is interested in knowing the number of pages of written math homework you complete in a day.

Problem 1

What is the interval of interest?

Solution

One day

Problem 2

What is the average number of pages you should do in one day?

Solution

2

Problem 3

Let XX = ____________. What values does XX take on?

Solution

Let XX = the number of pages of written math homework you do per day.

Problem 4

The probability question is P(______)P(______).

Solution

P(X > 2)P(X > 2)

Notation for the Poisson: P = Poisson Probability Distribution Function

X X ~ P(μ) P(μ)

Read this as "XX is a random variable with a Poisson distribution." The parameter is μμ (or λλ). μμ (or λλ) = the mean for the interval of interest.

Example 4

Leah's answering machine receives about 6 telephone calls between 8 a.m. and 10 a.m. What is the probability that Leah receives more than 1 call in the next 15 minutes?

Let XX = the number of calls Leah receives in 15 minutes. (The interval of interest is 15 minutes or 1414 hour.)

XX takes on the values 0, 1, 2, 3, ...

If Leah receives, on the average, 6 telephone calls in 2 hours, and there are eight 15 minutes intervals in 2 hours, then Leah receives

18 ⋅6 =0.75 18 ⋅6 0.75

calls in 15 minutes, on the average. So, μμ = 0.75 for this problem.

XX ~ P(0.75)P(0.75)

Find P(X>1) P(X 1). P(X>1)=0.1734 P(X 1) 0.1734 (calculator or computer)

TI-83+ and TI-84: For a general discussion, see this example (Binomial). The syntax is similar. The Poisson parameter list is (μμ for the interval of interest, number). For this problem:

Press 1- and then press 2nd DISTR. Arrow down to C:poissoncdf. Press ENTER. Enter .75,1). The result is P(X>1)=0.1734 P(X 1) 0.1734. NOTE: The TI calculators use λλ (lambda) for the mean.

The probability that Leah receives more than 1 telephone call in the next fifteen minutes is about 0.1734.

The graph of XX ~ P(0.75)P(0.75) is:

The poisson probability distribution function graph has 5 bars decreasing from left to right with an x-axis of 0-∞ and a y-axis of 0-0.5 in increments of 0.1. The x-axis is equal to the number of calls Leah receives within 15 minutes.

The y-axis contains the probability of XX where XX = the number of calls in 15 minutes.

Glossary

Poisson Distribution:
A discrete random variable (RV) that counts the number of times a certain event will occur in a specific interval. Characteristics of the variable:
  • The probability that the event occurs in a given interval is the same for all intervals.
  • The events occur with a known mean and independently of the time since the last event.
The distribution is defined by the mean μμ of the event in the interval. Notation: X~P(μ)X~P(μ) size 12{X "~" P \( μ \) } {}. The mean is μ=npμ=np size 12{μ= ital "np"} {}. The standard deviation is σ = μ σ=μ. The probability of having exactly xx successes in rr trials is P(X=x)=e−μμxx!P(X=x)=e−μμxx! size 12{P \( X=x \) =e rSup { size 8{ - μ} } { {μ rSup { size 8{x} } } over {x!} } } {}. The Poisson distribution is often used to approximate the binomial distribution when nn is “large” and pp is “small” (a general rule is that nn should be greater than or equal to 20 and pp should be less than or equal to .05).

Content actions

Give Feedback:

E-mail the module authors | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks