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**Unformatted text preview: **|^,Mk € ^rh Stat 400 Itrture 8 J vr*k5 ba,$n V- t) Review (2.5) o Moment-generating function (r.g.f.) and its proper- ties o Calculating mean and variance through m.g.f. o A note about discrete distribution Today's Lecture(2.6) o Poisson Processes, Poisson distribution. o Mean, variance and m.g.f. of Poisson distribution. o Poisson Approximation to Binomial distribution. St&t 400 [nture 8 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 r 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. Defini,tion: The random variable X has a Po'isson d;is- tributi,on if its p.m.f. is of the form l@) : #."-^, for r : 0,t,2,... where .\ > 0. 20t2 Vo{l++++ > a l*-,"sa eJ -e)-rt\ f r/ t ./t ,&,) X:-l lof10 Stst 400 lEcture 8 Connections between Poisson process and Poisson distribution Let Xt be the number of events to occur in time t (units). Then Xr - Poisson(.\t), and (.\r), P(Xt : ,) : ";i....

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