# X x theorem 0 1 x x proof 0 0 0 1 x x x x x x e e e e

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xx.Theorem:0( )1xx.Proof:000( )1!!xxxxxxeeeexxTheorem:The mean of the Poisson distribution isand the variance of a Poisson distribution is.Proof:10110[ ]!(1)!(1)!( )!xxxxxxxxE xxeeeexxxxA similar procedure shows2[ ]E x. Thus,22[ ]( )Var x.For certain values of, CDF charts are provided. Using these CDF charts are very straight forward afteridentifying. Using the Poisson CDF chart: The Poisson CDF chart is actually several different CDFcharts with a selection of possible rates (). First identifyfor the problem and use that column toanswer your questions. The value ofcould be a non-integer. You can have an arrival rate of 7.6 perhour. Since7.6is not on our charts, we would have to use the Poisson distribution formula todetermine our probabilities. In the real world you would use a computer program instead of thesecharts or the formula.
24If9and we wanted to determine(8)P X, we would go to the appropriate column and look downto8xand finalize our answer:(8)(8).4557P X Example:Determine(6)P Xfor a Poisson random variable with9(6)(6)(5).2068.1157.0909P X .
Exercise:Determine(7)PXfor a Poisson random variable with4Theorem:If the number of events in a time interval t follows a Poisson distribution with mean, thenthe number of events in time periodk tfollows a Poisson distribution with meankThis is quite a common sense theorem. If arrivals occur at a rate of 10 per hour, we could also say 20 per2-hour period, 5 per half hour, 2.5 per 15-minute period etc.Suppose thatXcounts the number of arrivals in a 20 minute period andPoi(8)X. (ReadXdistributed). Thus, the rate is 8 per 20-minute time period. Suppose that we want to determineprobability questions over a 25-minute period. What would our newbe?..is,
Exercise:Determine(9)PY.
25Exercise:The number of patients requiring a certain type of treatment in a given day well modeled by a Poissondistribution with a mean rate of6. The facility currently has 4 machines that are required for the treatment.Each machine can be used on 2 patients per day.a) Determine the probability that they have enough machines to treat all patients on a given day.b) How many machines would they need if they wanted the probability that they would have enough machines totreat all patients on a given day to exceed .99?Exploratory Example:A person goes out into the hallway and tosses a coin every 10 seconds. Each timea heads is tossed, the person dings a bell. LetXcount the number of dings per minute. Clearly X has abinomial distribution with parameters6nand.5p. Is this a Poisson distribution too? All threecriteria seem to be met. We have a constant arrival rate of three dings per minute. We certainly haveindependent increments. There is no way to get two dings in a very short period of time. Let’s compare

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Term
Fall
Professor
Kuhlmann
Tags
Probability theory, Binomial distribution
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