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Stat400Lec11(Ch3.5,3.6) - Stat 400 Lecture 11 Spring 2012...

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Stat 400 Lecture 11 Spring 2012 Review (3.4,3.7) Uniform distribution, its mean, variance, m.g.f. Exponential distribution and its mean, variance. Memoryless properties of exponential distribution. Cauchy distribution, GOT A MOMENT? Today’s Lecture (3.5,3.6) Gamma distribution and its connection to Poisson Process Mean, variance and m.g.f. of Gamma distribution. χ 2 distribution. Normal (Gaussian) distribution 1 of 9
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Stat 400 Lecture 11 Spring 2012 Volcano eruption (again) Let N t be the number of volcano eruptions to have oc- curred by time t , starting from now. Suppose that the volcano eruption forms a Poisson process with rate λ . Then N t . Let X be the waiting time until the 4 -th volcano eruption occurs and find the distribution of X . F X ( x ) = In general, if X is the waiting time until the -th volcano eruption, then F X ( x ) = 1 - - 1 X k =0 ( λ x ) k e - λ x k ! , x > 0 and f X ( x ) = F 0 X ( x ) = λ x - 1 ( - 1)! e - λ x , x > 0 2 of 9
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Stat 400 Lecture 11 Spring 2012 Gamma function Γ ( t ) = Z 1 0 y t - 1 e - y dy, t > 0 Properties of Γ ( t ) :
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