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An introduction to probability theory concepts, including central moments, covariance, correlation, and continuous random variables. Central moments are defined and the significance of the second central moment (variance) is explained. Covariance and correlation coefficients are introduced as measures of the relationship between random variables. The document also covers continuous random variables and their probability density functions, cumulative distribution functions, and expectations.

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2011/2012

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Download Probability Theory: Moments, Covariance, Correlation, and Continuous Variables and more Slides Computational Physics in PDF only on Docsity! 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 Covariance & Correlation The covariance is a measure of the independence of two random variables x and y: yxyxyx },cov{ Zero covariance does not imply independence of random variables. Another quantity related to covariance is the correlation coefficient: It is equal to zero when x and y are independent. Also, }var{ },cov{ 22 x xxxx }var{}var{ },cov{ ),( yx yx yx Its value is in between -1 and +1. Monte Carlo calculations try to take advantage of the negative correlation as a measure of reducing the variance. 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 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 F(x) x 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 F'(x) x Exponential Probability Distribution Function The average value: The variance of x is 0,0 0),exp(1)( x xxxF 0,0 0),exp()( x xxxF /1)exp( )( dxxx dxxfxx 2 2 2 /1 )()(}var{ dxxfxdxxfxx docsity.com