Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Collaborative Statistics: Custom Version modified by V Moyle » Geometric Distribution

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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.
  • College Open Textbooks display tagshide tags

    This module is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook CollaborativeAs a part of collection: "Collaborative Statistics"

    Comments:

    "Reviewer's Comments: 'I recommend this book. Overall, the chapters are very readable and the material presented is consistent and appropriate for the course. A wide range of exercises introduces […]"

    Click the "College Open Textbooks" link to see all content they endorse.

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

  • JVLA Endorsed

    This module is included inLens: Jesuit Virtual Learning Academy Endorsed Material
    By: Jesuit Virtual Learning AcademyAs a part of collection: "Collaborative Statistics"

    Comments:

    "This is a robust collection (textbook) approved by the College Board as a resource for the teaching of AP Statistics. "

    Click the "JVLA Endorsed" link to see all content they endorse.

  • WebAssign display tagshide tags

    This module is included inLens: WebAssign The Independent Online Homework and Assessment Solution
    By: WebAssignAs a part of collection: "Collaborative Statistics"

    Comments:

    "Online homework and assessment available from WebAssign."

    Click the "WebAssign" 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.

  • Bookshare

    This module is included inLens: Bookshare's Lens
    By: Bookshare - A Benetech InitiativeAs a part of collection: "Collaborative Statistics"

    Comments:

    "DAISY and BRF versions of this collection are available."

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

  • 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

  • statistics display tagshide tags

    This module is included inLens: Statistics
    By: Brylie OxleyAs a part of collection: "Collaborative Statistics"

    Click the "statistics" link to see all content selected in this lens.

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

  • 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.

  • statf12

    This module is included inLens: Statistics Fall 2012
    By: Alex KolesnikAs a part of collection: "Collaborative Statistics"

    Click the "statf12" link to see all content selected in this lens.

  • UTEP display tagshide tags

    This module is included inLens: Amy Wagler's Lens
    By: Amy WaglerAs a part of collection: "Collaborative Statistics"

    Click the "UTEP" link to see all content selected in this lens.

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

  • Make Textbooks Affordable

    This module is included inLens: Make Textbooks Affordable
    By: Nicole AllenAs a part of collection: "Collaborative Statistics"

    Click the "Make Textbooks Affordable" link to see all content selected in this lens.

  • BUS204 Homework display tagshide tags

    This module is included inLens: Saylor BUS 204 Homework
    By: David BourgeoisAs a part of collection: "Collaborative Statistics"

    Comments:

    "Homework for Discrete Variables/Probability. "

    Click the "BUS204 Homework" link to see all content selected in this lens.

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

  • crowe

    This module is included in aLens by: Chris RoweAs a part of collection: "Collaborative Statistics"

    Click the "crowe" link to see all content selected in this lens.

  • 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.
 

Geometric Distribution

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

Summary: This module describes the geometric experiment and the geometric probability distribution. This module is included in the Collaborative Statistics textbook/collection as an optional lesson.

The characteristics of a geometric experiment are:

  1. There are one or more Bernoulli trials with all failures except the last one, which is a success. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bull's eye until you hit the bull's eye. The first time you hit the bull's eye is a "success" so you stop throwing the dart. It might take you 6 tries until you hit the bull's eye. You can think of the trials as failure, failure, failure, failure, failure, success. STOP.
  2. In theory, the number of trials could go on forever. There must be at least one trial.
  3. The probability,pp, of a success and the probability, qq, of a failure is the same for each trial. p+q=1 p+q 1 and q=1-p q 1-p . For example, the probability of rolling a 3 when you throw one fair die is 1616. This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first 3 on the fifth roll. On rolls 1, 2, 3, and 4, you do not get a face with a 3. The probability for each of rolls 1, 2, 3, and 4 is q=56 q 56 , the probability of a failure. The probability of getting a 3 on the fifth roll is 56 ⋅ 56 ⋅ 56 ⋅ 56 ⋅ 16 =0.0804 56 ⋅ 56 ⋅ 56 ⋅ 56 ⋅ 16 0.0804
X=X= the number of independent trials until the first success. The mean and variance are in the summary in this chapter.

Example 1

You play a game of chance that you can either win or lose (there are no other possibilities) until you lose. Your probability of losing is p = 0.57p = 0.57. What is the probability that it takes 5 games until you lose? Let XX = the number of games you play until you lose (includes the losing game). Then XX takes on the values 1, 2, 3, ... (could go on indefinitely). The probability question is P(x=5) P(x 5) .

Example 2

A safety engineer feels that 35% of all industrial accidents in her plant are caused by failure of employees to follow instructions. She decides to look at the accident reports (selected randomly and replaced in the pile after reading) until she finds one that shows an accident caused by failure of employees to follow instructions. On the average, how many reports would the safety engineer expect to look at until she finds a report showing an accident caused by employee failure to follow instructions? What is the probability that the safety engineer will have to examine at least 3 reports until she finds a report showing an accident caused by employee failure to follow instructions?

Let XX = the number of accidents the safety engineer must examine until she finds a report showing an accident caused by employee failure to follow instructions. XX takes on the values 1, 2, 3, .... The first question asks you to find the expected value or the mean. The second question asks you to find P(x≥3) P(x 3) . ("At least" translates as a "greater than or equal to" symbol).

Example 3

Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55% of the 25,000 students do live within five miles of you. You randomly contact students from the college until one says he/she lives within five miles of you. What is the probability that you need to contact four people?

This is a geometric problem because you may have a number of failures before you have the one success you desire. Also, the probability of a success stays the same each time you ask a student if he/she lives within five miles of you. There is no definite number of trials (number of times you ask a student).

Problem 1

Let XX = the number of ____________ you must ask ____________ one says yes.

Solution

Let XX = the number of students you must ask until one says yes.

Problem 2

What values does XX take on?

Solution

1, 2, 3, …, (total number of students)

Problem 3

What are pp and qq?

Solution

  • pp = 0.55
  • qq = 0.45

Problem 4

The probability question is P(_______).

Solution

P ( x  =  4 )P(x = 4)

Notation for the Geometric: G = Geometric Probability Distribution Function

XX ~ G(p)G(p)

Read this as "XX is a random variable with a geometric distribution." The parameter is pp. pp = the probability of a success for each trial.

Example 4

Assume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the 7th component tested. How many components do you expect to test until one is found to be defective?

Let XX = the number of computer components tested until the first defect is found.

XX takes on the values 1, 2, 3, ... where p = 0.02p= 0.02. XX ~ G(0.02)G(0.02)

Find P(x=7) P(x 7) . P(x=7)=0.0177 P(x 7) 0.0177 . (calculator or computer)

TI-83+ and TI-84: For a general discussion, see this example (binomial). The syntax is similar. The geometric parameter list is (p, number) If "number" is left out, the result is the geometric probability table. For this problem: After you are in 2nd DISTR, arrow down to D:geometpdf. Press ENTER. Enter .02,7). The result is P(x=7)=0.0177 P(x 7) 0.0177 .

The probability that the 7th component is the first defect is 0.0177.

The graph of XX ~ G(0.02)G(0.02) is:

The geometric probability distribution function graph consists of bars sloping downwards with each successive bar to the right. The x-axis is from 1-∞ and the y-axis is from 0-0.02 in increments of 0.005. The x-axis is equal to the number of computer components tested until the defect is found.

The yy-axis contains the probability of xx, where XX = the number of computer components tested.

The number of components that you would expect to test until you find the first defective one is the mean, μμ = 50.

The formula for the mean is μ=1p=10.02=50 μ 1p 10.02 50

The formula for the variance is σ2= 1p ⋅ ( 1p - 1 ) = 10.02 ⋅ ( 10.02 - 1 ) =2450 σ2 1p ⋅ ( 1p - 1 ) 10.02 ⋅ ( 10.02 - 1 ) 2450

The standard deviation is σ= 1p ⋅ ( 1p - 1 ) = 10.02 ⋅ ( 10.02 - 1 ) = 49.5 σ 1p ⋅ ( 1p - 1 ) 10.02 ⋅ ( 10.02 - 1 ) 49.5

Glossary

Geometric Distribution:
A discrete random variable (RV) which arises from the Bernoulli trials. The trials are repeated until the first success. The geometric variable XX is defined as the number of trials until the first success. Notation: XX∼ G ( p )G(p). The mean is μ = 1 p μ= 1 p and the standard deviation is σ=1p⋅(1p−1)σ=1p⋅(1p−1) The probability of exactly x failures before the first success is given by the formula: P(X=x)=p(1−p)x−1P(X=x)=p(1−p)x−1 size 12{P \( X=x \) =p \( 1 - p \) rSup { size 8{x - 1} } } {}.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

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

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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

Module to:

My Favorites (?)

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

| A lens I own (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to 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 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