Lecture 5: Probability Distribution, Moments, Variance, Covariance, and Correlation with N, Lecture notes of Mathematical Modeling and Simulation

Download Lecture 5: Probability Distribution, Moments, Variance, Covariance, and Correlation with N and more Lecture notes Mathematical Modeling and Simulation in PDF only on Docsity! Lecture 5: Nasir M Mirza 1 6/23/2012 Monte Carlo Simulation Room A-114, Department of Physics & Applied Mathematics, Pakistan Institute of Engineering & Applied Sciences, P.O. Nilore, Islamabad email: [email protected] Dr. Nasir M Mirza Lecture Five: Probability Distribution function Docsity.com Lecture 5: Nasir M Mirza 2 6/23/2012 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 Lecture 5: Nasir M Mirza 5 6/23/2012 Consider two events E0 and E1 that are mutually exclusive and exhaustive: Binomial Probability Distribution Function .0,0}0( .,1,}1{   xEP xpEP The expected values for the real number x and its square are .)( ,0)1(1)( 2 pxE pppxE   The variance of x is Suppose there are N independent samples of these events And each has either 0 or 1 outcome. Then probability of x Successes out of N is ).1( }var{ 2 22 pppp xxx   xNx x N ppCxXP  )1(}{ Npx  )1(}var{ pnpx The variance of x is The average or mean of x is Binomial pdf: Docsity.com Lecture 5: Nasir M Mirza 6 6/23/2012 Geometric pdf Suppose we have to carry out a certain experiment repeatedly and independently where there are only two outcomes failure or success. If the outcome is failure then the experiment is repeated; otherwise we stop. Now x is the number of experiments until success appears. Then ,,3,2,1,}{ 1   npqxXP x Where, q is the probability of failure in one experiment and p is the probability of success in one experiment. 2 1 1 )1( q p pqxx x x       The variance of x is pp x 11 }var{ 2  Docsity.com Lecture 5: Nasir M Mirza 7 6/23/2012 The Poisson Distribution A random variable x is said to follow Poisson distribution, when Where, l is a parameter of this distribution. It is easy to find that l x This distribution is fundamental in the theory of probability and stochastic processes. It is of great use in applications such as radioactive decay, queuing service systems and similar systems. l l  }var{x ,,3,2,1,0, ! )( }{   n n t exXP n t ll Docsity.com