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written by Harvey Gould and Jan Tobochnik
We introduce the basic concepts of probability and apply them to simple physical systems and everyday life. We will discover the universal nature of the central limit theorem and the Gaussian distribution for the sum of a large number of random variables and discuss its relation to why thermodynamics is possible. Because of the importance of probability in many contexts and the relatively little time it will take us to consider more advanced topics, our discussion goes beyond what we will need for the applications of statistical mechanics. Computer simulations will be used to explore these concepts. These simulations can be found by searching ComPADRE for Open Source Physics, STP, or Statistical and Thermal Physics.
Book Title: Thermal and Statistical Physics
Subjects Levels Resource Types
Mathematical Tools
- Probability
Thermo & Stat Mech
= Central Limit Theorem
= Gaussian Distribution
- Upper Undergraduate
- Instructional Material
= Textbook
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- Educators
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Access Rights:
Free access
Restriction:
© 2008 Harvey Gould and Jan Tobochnik
All chapters are copyrighted by Harvey Gould and Jan Tobochnik. They are not to be copied or distributed without contacting one of the authors.
PACSs:
02.50.-r
05.10.-a
Keywords:
Open Source Physics, osp
Record Creator:
Metadata instance created May 27, 2008 by Anne Cox
Record Updated:
June 6, 2014 by Harvey Gould
Last Update
when Cataloged:
May 1, 2008
Other Collections:

On the Binomial --> Poisson convergence

Author: jean-michel levy
Posted: February 2, 2010 at 10:35AM

Using Stirlin's formula here is akin to smash a fly with a hammer.
It is enough to observe that the binomial coefficient C_n^k can be rewritten:                                                            

n^k/k! [1(1-1/n)(1-2/n)..(1-(n-k)/n)]  The number of factors between brackets is fixed and they all tend to 1
when n--> infinity.

Replacing now p by l/n yields  P[X=k] ~ n^k/k! (l/n)^k(1-l/n)^{n-k} and the last factor --> exp(-l) in the
desired limit,  which completes the proof.

» reply

Re: On the Binomial --> Poisson convergence

Author: Harvey Gould
Posted: Feb 18, 2010 at 10:19PM

We agree and updated our explanation. Thank you.

Harvey Gould

> On Feb 02, 2010, jean-michel levy posted:
>
> Using Stirlin's
> formula here is akin to smash a fly with a hammer
>
> It is enough to observe that the binomial coefficient
> C_n^k can be rewritten:                          
>                                  
>
> n^k/k! [1(1-1/n)(1-2/n)..(1-(n-k)/n)]
>  The number of factors between brackets is fixed and
> they all tend to 1
> when n--> infinity.
>
> Replacing
> now p by l/n yields  P[X=k] ~ n^k/k! (l/n)^k(1-l/n)^{n-k}
> and the last factor --> exp(-l) in the
> desired limit,
>  which completes the proof.

» reply

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AIP Format
H. Gould and J. Tobochnik, in Thermal and Statistical Physics (2008), WWW Document, (http://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446).
AJP/PRST-PER
H. Gould and J. Tobochnik, STP Textbook Chapter 3: Concepts of Probability, in Thermal and Statistical Physics (2008), <http://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446>.
APA Format
Gould, H., & Tobochnik, J. (2008). STP Textbook Chapter 3: Concepts of Probability. In Thermal and Statistical Physics. Retrieved December 22, 2014, from http://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446
Chicago Format
Gould, Harvey, and Jan Tobochnik. "STP Textbook Chapter 3: Concepts of Probability." In Thermal and Statistical Physics. 2008. http://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446 (accessed 22 December 2014).
MLA Format
Gould, Harvey, and Jan Tobochnik. "STP Textbook Chapter 3: Concepts of Probability." Thermal and Statistical Physics. 2008. 1 May 2008. 22 Dec. 2014 <http://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446>.
BibTeX Export Format
@incollection{ Author = "Harvey Gould and Jan Tobochnik", Title = {STP Textbook Chapter 3: Concepts of Probability}, BookTitle = {Thermal and Statistical Physics}, Month = {May}, Year = {2008} }
Refer Export Format

%A Harvey Gould
%A Jan Tobochnik
%T STP Textbook Chapter 3: Concepts of Probability
%B Thermal and Statistical Physics
%D May 1, 2008
%U http://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446
%O application/pdf

EndNote Export Format

%0 Book Section
%A Gould, Harvey
%A Tobochnik, Jan
%D May 1, 2008
%T STP Textbook Chapter 3: Concepts of Probability
%B Thermal and Statistical Physics
%8 May 1, 2008
%U http://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446


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STP Textbook Chapter 3: Concepts of Probability:

Is Referenced By STP Binomial Program

STP Binomial Program illustrates concepts from Chapter 3 of Statistical and Thermal Physics by Gould and Tobochnik.

relation by Anne Cox
Is Part Of STP Textbook: Table of Contents

The STP Textbook is an introduction to statistical and thermal physics for upper-level undergraduate students.

relation by Anne Cox
Is Referenced By STP Coin Toss Program

STP Coin Toss Program illustrates concepts from Chapter 3 of Statistical and Thermal Physics by Gould and Tobochnik.

relation by Anne Cox
Is Referenced By STP Central Limit Theorem Program

STP Central LImit Theorem Program illustrates concepts from Chapter 3 of Statistical and Thermal Physics by Gould and Tobochnik.

relation by Anne Cox
Is Referenced By STP Random Walk 1D Continuous Program

STP Random Walk 1D Continuous Program illustrates concepts from Chapter 3 of Statistical and Thermal Physics by Gould and Tobochnik.

relation by Anne Cox

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