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An in-depth exploration of continuous probability distributions in the context of statistical physics. Topics covered include moments and variance, covariance and correlation, and the calculation of expectations for continuous random variables. Real-life examples of uniform, exponential, and normal distributions are also included, along with matlab code for generating their probability density functions (pdf) and cumulative distribution functions (cdf).

Typology: Slides

2011/2012

1 / 10

Download Understanding Moments, Covariance, and Correlation in Continuous Probability Distributions and more Slides Statistical Physics in PDF only on Docsity! Chapter 2 Topic: continuous pdf Simulations in Statistical Physics Docsity.com Moments & Variance The Central Moments The Central moments of x are defined as The second central moment has a particular meaning: This is also called variance of x. n i n xxp xxxg )( )()( 22 22 22 )()( xx xxp xxpxx i i i i i i i 22}var{ xxx }var{xThe standard deviation of x is Docsity.com % How to generate uniform dis. % The domain is generated x = -1:0.1:11; % now the pdf for x values pdf = unifpdf(x, 0, 10); cdf = unifcdf(x, 0, 10); subplot(1,2,1),plot(x,pdf) title('pdf') xlabel('X'), ylabel('f(x)') axis([-1 11 0 0.2]) axis square subplot(1,2,2),plot(x,cdf) title('cdf') xlabel('X'), ylabel('f(x)') axis([-1 11 0 1.1]) axis square MATLAB Program to Generate Uniform pdf & cdf Docsity.com Expectations of Continuous Random Variables The mean value of a continuous random variable in an interval [a, b] is Where, f(x) is the probability density function (pdf) for x. The normalization condition is The expected and variance value of any function of g(x) with this pdf are b a b a dxxfx xxdFxE )( )()( 22 )()(}var{ gEgEg )(1)( Fdxxf dxxfxggE b a )()()( Docsity.com Examples of Continuous Probability Distributions UNIFORM DISTRIBUTION: ax axx xxF ,1 0, 0,0)( Average: 2/ax Brief Calculations: ax axa xxF ,0 0,/1 0,0)( Variance: 12/}var{ 2ax 124 )( }var{ 22 0 2 22 aa dxxfx xxx a 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 III II F (x ) x 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 F '( x) x Docsity.com