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# INTR THY PROBAB&S I STAT 341

ISU

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This 2 page Class Notes was uploaded by Giovani Ullrich PhD on Saturday September 26, 2015. The Class Notes belongs to STAT 341 at Iowa State University taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/214406/stat-341-iowa-state-university in Statistics at Iowa State University.

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Date Created: 09/26/15

Poisson Distribution The Poisson Distribution is used to model the number of rare events that occur in space time volume or any other dimension For this distribution The average number of events in a given time period or space is o The number of events in non overlapping time periods or spaces are independent The probability of one event in a short time period or space h is h The probability of more than one event in a short time period or space h is O The random variable Y number of events in a given time period or space The parameter for the Poisson random variable Y is The probability distribution function for the Poisson random variable Y is Aye A yl y012 The theoretical mean of the Poisson random variable Y is EY The theoretical variance of the Poisson random variable Y is VY A Poisson distribution can be used to model the number of accidents that occur within a week at a given intersection the number of telephone calls handled by a switchboard in a given time interval the number of radioactive particles that decay in a particular time period the number of errors a typist makes in typing a page and the number of automobiles using a freeway access ramp in a 10 minute interval Unlike the other discrete distributions we ve studied we must either assume the random variable Y has a Poisson distribution usually based on the information given above or we must check that the Poisson distribution would seem a reasonable model to use based on an analysis of collected data Working with Poisson random variables in R To nd a probability PY y py for a single value y the command in R is dpois y lambda To nd the probability PY y use the sum command to add up all py values for y between and including 0 and y sumdpois O y lambda To nd the probability Py1 g Y yg use the sum command to add up all py values for y between and including yl and yg sumdpoisy1y2 lambda To nd the probability PY 2 y 17 PY lt y 17 PY y 7 17 use the sum command to nd PY y 7 1 and subtract this value from 1 1 sumdpois0y1 lambda To generate values from a Poisson random variable Y with mean 7 the command in R is rpois numobs lambda where numobs is the number of observations of the random variable Y you would like to generate 1 2 a Problems Show that py has the two properties of a probability distribution function Derive the expected value of a Poisson random variable Use R to generate 10000 observations from a Poisson random variable with 3 Make a histogram of your data and calculate the mean7 variance7 and ve number summary of your observations Use this information to describe the distribution of your data Now use R to generate 10000 observations from a Poisson random variable with 10 Make a histogram of your data and calculate the mean7 variance7 and ve number summary of your observations How does this distribution compare with the one from problem 3 A police of ce visits a location y 07 17 27 37 times every half hour On average7 the police of cer visits a location once every half hour For the next half hour time period7 nd the probability the police of cer will miss a particular location visit the location once Tree seedlings are randomly dispersed in a particular area with an average dispersement of 5 seedlings per square yard a In a one square yard area7 what is the probability that no seedlings are found b In ten one square yard areas7 what is the probability that no seedlings are found c In ten one square yard areas7 what is the probability that at least one out of the ten areas contains seedlings d In ten one square yard area7 what is the probability that exactly three out of the ten areas contains seedlings

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