STP Textbook Chapter 3: Concepts of Probability

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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

download 580kb .pdf
Published: May 1, 2008
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Subjects	Levels	Resource Types
Mathematical Tools - Probability Thermo & Stat Mech = Central Limit Theorem = Gaussian Distribution	- Upper Undergraduate	- Instructional Material = Textbook

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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:: The Open Source Physics

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 10:19PM UTC

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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<a href="https://www.compadre.org/STP/items/detail.cfm?ID=7272">Gould, Harvey, and Jan Tobochnik. "STP Textbook Chapter 3: Concepts of Probability." In Thermal and Statistical Physics. 2008.</a>

AIP Format

H. Gould and J. Tobochnik, , in Thermal and Statistical Physics (2008), WWW Document, (https://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), <https://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 12, 2024, from https://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. https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446 (accessed 12 December 2024).

MLA Format

Gould, Harvey, and Jan Tobochnik. "STP Textbook Chapter 3: Concepts of Probability." Thermal and Statistical Physics. 2008. 1 May 2008. 12 Dec. 2024 <https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446>.

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@incollection{ Author = "Harvey Gould and Jan Tobochnik", Title = {STP Textbook Chapter 3: Concepts of Probability}, BookTitle = {Thermal and Statistical Physics}, Month = {May}, Year = {2008} }

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

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%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 https://www.compadre.org/Repository/document/ServeFile.cfm?ID=7272&DocID=446

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