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**Unformatted text preview: **Stat 400 Lecture 8 Spring 2012 Review (2.5) Moment-generating function (m.g.f.) and its proper- ties Calculating mean and variance through m.g.f. A note about discrete distribution Todays Lecture(2.6) Poisson Processes, Poisson distribution. Mean, variance and m.g.f. of Poisson distribution. Poisson Approximation to Binomial distribution. 1 of 10 Stat 400 Lecture 8 Spring 2012 Poisson distribution Examples: 1. Number of telephone calls arriving at a switchboard between 3pm and 5pm. 2. Number of defects in a 100-foot roll of aluminum screen that is 2 feet wide. 3. Number of road accidents in a year in US. Poisson Process The Poisson process counts the num- ber of events occurring in a fixed time or space, when the number of events occurring in non-overlapping intervals are independent. events occur at a constant average rate of per unit time. events cannot occur simultaneously. Definition: The random variable X has a Poisson dis- tribution if its p.m.f. is of the form f ( x ) = x x ! e- , for x = 0 , 1 , 2 , where > . 2 of 10 Stat 400 Lecture 8 Spring 2012 Connections between Poisson process and Poisson distribution Let X t be the number of events to occur in time t (units)....

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