Lectures 20/21 Poisson distributionBinomial and Poisson approximationAdvantage: No need to know n and p; estimate the parameter l from dataPowerPoint PresentationRutherfold and Geiger (1910)Slide 6Slide 7Slide 8Slide 9Lectures 20/21 Poisson distribution•As a limit to binomial when n is large and p is small.•A theorem by Simeon Denis Poisson(1781-1840). Parameter = np= expected value•As n is large and p is small, the binomial probability can be approximated by the Poisson probability function •P(X=x)= e- x / x! , where e =2.71828•Ion channel modeling : n=number of channels in cells and p is probability of opening for each channel;Binomial and Poisson approximationx n=100, p=.01 Poisson0 .366032 .3678791 .36973 .3678792 .184865 .1839403 .06099 .0613134 .014942 .0153285 .002898 .0030666 .0000463 .0005117Advantage: No need to know n and p; estimate the parameter from dataX= Number of deaths frequencies0 1091 652 223 34 1total 200200 yearly reports of death by horse-kick from10 cavalry corps over a period of 20 years in 19th century by Prussian officials.x Data frequenciesPoissonprobabilityExpected frequencies0 109 .5435 108.71 65 .3315 66.32 22 .101 20.23 3 .0205 4.14 1 .003 0.6200Pool the last two cells and conduct a chi-square test to see if Poisson model is compatible with data or not. Degree of freedom is 4-1-1 = 2. Pearson’s statistic = .304; P-value is .859 (you can only tell it is between .95 and .2 from table in the book); accept null hypothesis, data compatible with modelRutherfold and Geiger (1910)•Polonium source placed a short distance from a small screen. For each of 2608 eighth-minute intervals, they recorded the number of alpha particles impinging on the screenMedical Imaging : X-ray, PET scan (positron emission tomography), MRI Other related application in# of particlesObserved frequency Expected freq.0 57 541 203 2112 383 4073 525 5264 532 5085 408 3946 273 2547 139 1408 45 689 27 2910 10 1111+ 6 6Pearson’s chi-squared statistics = 12.955; d.f.=12-1-1=10 Poisson parameter = 3.87, P-value between .95 and .975. Accept null hypothesis : data are compatible with Poisson modelPoisson process for modeling number of event occurrences in a spatial or temporal domainHomogeneity : rate of occurrence is uniformIndependent occurrence in non-overlapping
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