Lecture_15

# Lecture_15 - 2 EXPONENTIAL AND GAMMA DISTRIBUTIONS Lecture...

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2 EXPONENTIAL AND GAMMA DISTRIBUTIONS Lecture 15 ORIE3500/5500 Summer2009 Chen Class Today Multivariate Normal (Gaussian) Distribution Exponential and Gamma Distributions 1 Multivariate Normal Distribution The bivariate normal distribution is one of the most important joint distrib- utions. ( X,Y ) are said to be a bivariate normal vector is they have density, for all , -∞ < x,y < , f X,Y ( x,y ) = 1 2 πσ 1 σ 2 p 1 - ρ 2 exp - ( x - μ 1 ) 2 σ 2 1 - 2 ρ ( x - μ 1 )( y - μ 2 ) σ 1 σ 2 + ( y - μ 2 ) 2 σ 2 2 2(1 - ρ 2 ) · . The parameters μ 1 2 1 2 and ρ , characterize the distribution. μ 1 and μ 2 are the means of X and Y respectively and σ 2 1 , σ 2 2 are the variance of X and Y respectively. ρ is the correlation between X and Y . We know that in general if two variables are independent then they have 0 correlation. But the other way is not true. That is , if two random variables are uncorrelated then they are not necessarily independent. But if ( X,Y ) have bivariate joint normal distribution and they are uncorrelated, then they are independent. This is a property of the normal distribution. The multivariate normal distribution is characterized by its mean and variance-covariance matrix. 2 Exponential and Gamma distributions Although the normal distribution ﬁnds widespread application in many areas, it can not be used to model everything. For example, the waiting time of a person in a queue, the service time of a customer in a counter, duration of 1

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Lecture_15 - 2 EXPONENTIAL AND GAMMA DISTRIBUTIONS Lecture...

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