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Unformatted text preview: 2 EXPONENTIAL AND GAMMA DISTRIBUTIONS Lecture 15 ORIE3500/5500 Summer2011 Li 1 Multivariate Normal Distribution The bivariate normal distribution is one of the most important joint distri- butions. ( 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 ,μ 2 ,σ 1 ,σ 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 a phone call etc. can not be modeled by a normal random variable for the simple fact that these are all nonnegative random variables. In some such cases, the exponential distribution or the gamma distribution can be used. A continuous random variable is said to follow exponential distribution with parameter λ , or X ∼ exp( λ ), if it has density f X ( x ) = λe- λx ,x ≥ . 1 2 EXPONENTIAL AND GAMMA DISTRIBUTIONS The cdf of X is F X ( x ) = 1- e- λx ,x ≥ . A random variable having an exp(1) distribution is said to be a standard exponential random variable. We compute the moment generating function of X ∼ exp( λ ) φ X ( t ) = Z ∞ e tx λe- λx dx = ‰ λ λ- t t < λ ∞ otherwise....
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- Spring '11
- Normal Distribution, Probability theory, GAMMA