Discrete Distributions: Poisson Distribution 1

Transcription

1 Discrete Distributions: Poisson Distribution 1 November 6, HMS, 2017, v1.1

2 Chapter References Diez: Chapter 3.3, 3.4 (not 3.4.2), Navidi, Chapter 4.1, 4.2, 4.3 Chapter References 2

3 Poisson Distribution The Poisson distribution describes the distribution of the number of independent events that occur in a given interval in time or space. Examples: 1. Number of cells per square mm on a Hemacytometer. 2. The number of mutations in set sized regions of a chromosome. 3. The number of particles emitted by a radioactive source in a given time. 4. The number of cases of a disease in different towns. Poisson Distribution 3

4 Poisson Distribution Events per unit time 1. Telephone called received in a hour 2. Articles received in a day at an airlines lost and found 3. Car accidents in a month at a busy intersection 4. Deaths per month due to a rare disease Events per unit distance 1. Defects occurring in 50 meters of insulated wire 2. Deaths per 10,000 passenger miles Events per unit area 1. Bacteria per square centimeter of culture plate Events per unit volume 1. White blood cells in a cubic millimeter of blood 2. Hydrogen atoms per cubic light-year in intergalactic space Poisson Distribution 4

5 Poisson Distribution The number of cars that pass under a road bridge during a given period of time. The number of spelling mistakes while typing a single page. The number of phone calls at a call center per minute. The number of times a web server is accessed per minute. The number of animals killed per unit length of road. Number of mutations per 100,000 base-pairs on DNA after a certain amount of radiation. The number of pine trees per unit area of mixed forest. The number of stars in a given volume of space. The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. The number of light bulbs that burn out in a certain amount of time. The number of viruses that can infect a cell in cell culture. The number of inventions invented over a span of time in an inventor s career. Number of particles that scatter off of a target in a nuclear experiment. The number of hurricanes in a year that originate in the Atlantic ocean. Poisson Distribution 5

6 Poisson Distribution Let X be the random variable that describes the number of occurrences of event in a given time, distance, area, volume etc. Given the following conditions: Events occur one at a time (no coincident events) - i.e events are rare Events are independent The probability of an event over a given period is constant You ll sometimes see the third condition expressed something like: The probability of observing a single event over a small interval is approximately proportional to the size of that interval. Poisson Distribution 6

7 Poisson Distribution Let X be the random variable that describes the number of occurrences of an event in a given time, distance, area, volume etc. Then X is distributed according to the Poisson distribution, given by: p(x) = λx e λ x! where λ is the mean number of events per interval. Note it only has a single parameter. Poisson Distribution 7

8 Poisson Distribution Mean: µ = λ σ 2 = λ Poisson Distribution 8

9 Poisson Distribution Poisson Distribution 9

10 Poisson Distribution Source: distribution The distribution becomes more symmetrical as λ increases. Assuming large n and small q such that np is constant, the Binomial distribution closely resembles a Poisson distribution. Poisson Distribution 10

11 Poisson Distribution: Example 1 If there are base pairs in the human genome and the mutation rate per generation per base pair is 10 9, what is the mean number of new mutations that a child (=genome) will have, what is the variance in this number, and what will the distribution look like? µ = λ σ 2 = λ Poisson Distribution 11

12 Poisson Distribution: Example 2 In a small US town the number of accidents per year is 2.4. a) What is the probability that in any particular year there will be no accidents? b) What is the probability that in any particular year there will be 5 accidents? Are the conditions met? Accidents only happen one at a time. We assume accidents are independent of each other The probability of the number of accidents per year remains the same (possibly not true). λ = 2.4 Poisson Distribution 12

13 Poisson Distribution: Example 2 a) What is the probability that in any particular year there will be no accidents? b) What is the probability that in any particular year there will be 5 accidents? λ = 2.4 p(x = 0) = e 2.4 (2.4 0 ) 0! p(x = 5) = e 2.4 (2.4 5 ) 5! = = 0.06 Poisson Distribution 13

14 Poisson Distribution: Example 2 Poisson Distribution 14

15 Poisson Distribution: Example 2 What is the probability that that there will be more than 4 accidents per year? Poisson Distribution 15

16 Poisson Distribution: Example 2 P (X > 4) = 1 P (X 4) P (X > 4) = 1 (p(0) + p(1) + p(2) + p(3) + p(4)) P (X > 4) = 1 (e e e + e + e 2! 3! ( ) P (X > 4) = 1 e ! 3! 4! P (X > 4) = ! ) Poisson Distribution 16

17 Poisson Distribution: Example 3 The mean number of white blood cells under a hemocytometer is 2. a) Probability of finding no cells P (x = 0) b) Probability of finding three cells, P (x = 3) c) Probability of finding greater than 2 cells, P (x > 2) Poisson Distribution 17

18 Poisson Distribution: Example 3 Are the conditions met? Events occur one at a time (no coincident events) Events are independent The probability of an event over a given period is constant Poisson Distribution 18

19 Poisson Distribution: Example 3 λ = 2 λ λ0 p(0) = e n! = e 2 = λ λ3 p(3) = e 3! = e 2 = P (x > 2) = 1 (p(0) + p(1) + p(2)) = Poisson Distribution 19

20 Poisson Distribution: Example 4 Rutherford, Geiger, and Bateman (1910) counted the number of α-particles emitted by a film of polonium in 2608 successive intervals of one-eighth of a minute. They counted 10,097 alpha particles. Number of Alpha Particles Observed in one-eighth of a minute Over 14 0 Total 2608 Poisson Distribution 20

21 Poisson Distribution: Example 4 For example, in 383 intervals they observed 2 particles. Number of Alpha Particles Observed in one-eighth of a minute Over 14 0 Total 2608 Poisson Distribution 21

22 Poisson Distribution: Example 4 What is the average number of particles observed per 7.5 seconds? The total number of α-particles in these 2608 periods is = The mean count per period is. λ = = 3.87 Poisson Distribution 22

23 Poisson Distribution: Example 4 Consider the Poisson distribution with parameter The following is its probability function x p(x) = e x! Poisson Distribution 23

24 Poisson Distribution: Example 4 For example, in 383 intervals they observed 2 particles. Number of Alpha Particles Observed in one-eighth of a minute Expected Over Total 2608 Poisson Distribution 24

25 Poisson Distribution: Example 5 The infection rate at a Neonatal Intensive Care Unit (NICU) is typically expressed as a number of infections per patient days. This is obviously counting a number of events across both time and patients. Does this data follow a Poisson distribution? Poisson Distribution 25

26 Poisson Distribution: Example 5 The infection rate at a Neonatal Intensive Care Unit (NICU) is typically expressed as a number of infections per patient days. This is obviously counting a number of events across both time and patients. Does this data follow a Poisson distribution? Independence: One child doesn t infect another child - unlikely Constant Probability: Likelihood of a child catching infection is not influence by length of stay - unlikely Poisson Distribution 26

27 Poisson Distribution: Example 5 A researcher counts the number of cars that pass by a busy street during one minute intervals. They compute a mean of 10.3, and a variance of only 5.3. For a Poisson distribution, the variance and mean should be the same. What is likely to have gone wrong? Poisson Distribution 27

28 Poisson Distribution: Example 6 A researcher counts the number of cars that pass by a busy street during one minute intervals. They computes a mean of 10.3, and a variance of only 5.3. For a Poisson distribution the variance and mean should be the same. What is likely to have gone wrong? Independence: Traffic lights etc might change spacing between cars, bunching or spread out. Constant Probability: Rush-hour and quiet hours? Poisson Distribution 28

29 Poisson Distribution: Example 7 A 5-litre bucket of water is taken from a swamp. The water contains 75 mosquito larvae. A 200 ml flask of water is taken form the bucket for further analysis. What is a) The expected number of larvae in the flask? b) The probability that the flask contains at least one mosquito lava? Poisson Distribution 29

30 Poisson Distribution: Example 7 A 5-litre bucket of water is taken from a swamp. The water contains 75 mosquito larvae. A 200 ml flask of water is taken form the bucket for further analysis. What is a) the expected number of larvae in the flask? Expected number: = 25 x = 75 3 = 3 b) The probability that the flask contains at least one mosquito larva? The question is asking what is the probability of getting 1 or 2 or 3 or... larva? 3 30 P (X 1) = 1 P (X = 0) = 1 e = = ! Poisson Distribution 30

31 Poisson Distribution: Example 7 Poisson Distribution 31

32 Poisson Distributions using Python import numpy as np from scipy.stats import poisson import pylab x = np.arange(-1, 200) q = 0.7 # create a Poisson distribution dist = poisson(q) pylab.vlines (x, 0, dist.pmf(x), color= r, linewidth=6) pylab.xlim(-0.5, 6); pylab.ylim(0, 0.6) pylab.xlabel( $x$ ) pylab.ylabel(r $p(x b, n)$ ) pylab.title( Poisson Distribution, n= + q = + str (q)) Poisson Distribution 32