Probability Theory: Moments, Covariance, Correlation, and Continuous Variables, Slides of Computational Physics

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{xThe 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