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1、Common Probability Distributions II8thweek/Probability and Statistics(II)Poisson Distribution ();Poisson random variable =/!,=0,1,2,0,otherwise =;Var=When events are observed on average,the probability to see events actually2Poisson Distribution events on average over time slots Prob.of event occurr
2、ence in each slot/:the number of actually observed events (,/)When ,(,/)()slots eventstime3Poisson Distribution=1051015200.000.050.100.150.304.00.350.250.20=4=10https:/en.wikipedia.org/wiki/Poisson_distribution4Exponential Distribution ();exponential random variable =for 0 =1;Var=12 Waiting time for
3、 an event when events occur on average in a unit time5Exponential Distribution ()Let Kbe a discrete RV s.t.Pr =Pr 1 Then,=1=(1 )()1 (1 )Geometric:waiting time for the first success Exponential:waiting time for an event6Exponential Distribution=0.5012340.00.20.40.61.21.61.41.00.8=1=1.55=0.5012340.00.
4、20.40.61.00.8=1=1.55 https:/en.wikipedia.org/wiki/Exponential_distribution7Gaussian(Normal)Distribution (,2);Gaussian(normal)random variable =122222,=;Var=2 Stanford normal distribution (0,1),=Pr +,2 0,18Gaussian(Normal)Distribution,2012340.00.20.40.61.00.85054321,2012340.00.20.40.61.00.85054321=0,2
5、=0.2=0,2=1.0=0,2=5.0=2,2=0.5=0,2=0.2=0,2=1.0=0,2=5.0=2,2=0.5https:/en.wikipedia.org/wiki/Normal_distribution9Gaussian(Normal)Distribution Central limit theoremY=1+2+are identical and independent,Y Normal distribution Real phenomenon is often modeled by the aggregation of many small and independent e
6、vents So,normal distribution is important for the real applications10Gaussian(Normal)Distribution Outcome of dices:K=1+2+https:/en.wikipedia.org/wiki/Normal_distribution11SummaryPMFExpVarDescription(),=11 ,=0(1 )Yes or No,Success or fail(1 )11/(1 )/2#of Bernoulli trials up to the first success,(1 )(
7、1 )#of successes among Bernoulli trails,+1(1 )(1 )(1 )2#of successes to see r failures in Bernoulli trials()/!,#of events,1 +1+/2()(+2)12Equiprobable12SummaryPMFExpVarDescription,1(),0,otherwise+2()212Equiprobable 1/1/2Waiting time for an event ,21222222Central limit theorem13ReferencesProbability and Stochastic Processes:A Friendly Introduction to Electrical and Computer Engineers(3rd edition),Yates and Goodman,WileyProbability,Statistics,and Random Processes for Electrical Engineering(3rd edition),Leon-Garcia,Pearson International Edition.