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Distribution Gamma Function Stochastic Process Tutorial 4, STAT1301 Fall 2010, 12OCT2010, MB103@HKU By Joseph Dong

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Reference 2

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Recall: Distribution of a Random Variable 3

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Handout Problems 6 & 7 Problem 6: ▫Gamma function and integration practice Problem 7: ▫important continuous distributions and their relationships 6 Technical

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From Bernoulli Trials to Discrete Waiting Time (Handout Problems 1-4) 7

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Poisson [ pwa’s ɔ ̃] Distribution Poisson Approximation to Binomial (PAB) ▫Handout Problem 5 The true utility of Poisson distribution—Poisson process: ▫Sort of the limiting case of Bernoulli trials (use PAB to facilitate thinking) ▫“continuous” Bernoulli trials 8

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Sequence of Random Variables 9

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Stochastic Process: Discrete-time and Continuous-time 10

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Discrete-time processContinuous-time process 11

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Bernoulli Trials (Bernoulli Process) 12

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Poisson Process 13

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Discrete Distribution Based On Bernoulli Trails 14

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Continuous Distribution Based On Poisson Process 15

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Examples of Poisson Process Radioactive disintegrations Flying-bomb hits on London Chromosome interchanges in cells Connection to wrong number Bacteria and blood counts Feller: An Introduction to Probability Theory and Its Applications (3e) Vol. 1. §VI.6. 16

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Radioactive Disintegrations 17 Geiger Counter Geiger Rutherford Chadwick

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Rutherford, Chadwick, and Ellis’ 1920 Experiment #intervals (recorded) 05754.5439956 1203210.9397008 2383407.8868524 3525525.8111954 4532508.371522 5408393.2082186 6273253.4444078 7139140.0219269 84567.68889735 92729.08615451 1616.99712879 N 2608 Intensity (7.5s) 3.867331 18

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Explanation There are 57 time intervals (7.5 sec each) recorded zero emission. There are 203 time intervals (7.5 sec each) recorded 1 emission. …… There are total 2608 time intervals (7.5 sec each) involved. On average, each interval recorded 3.87 emissions. Use 3.87 as the intensity of the Poisson process that models the counts of emissions on each of the 2608 intervals. 19

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What’s the waiting time until recording 40 emissions? 20

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