Poisson Distribution<br />Given a Poisson process, the probability of obtaining exactly successes in trials is given by th...
Poisson distribution jen
Poisson distribution jen
Poisson distribution jen
Poisson distribution jen
Poisson distribution jen
Poisson distribution jen
Poisson distribution jen
Poisson distribution jen
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Poisson distribution jen

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Poisson distribution jen

  1. 1. Poisson Distribution<br />Given a Poisson process, the probability of obtaining exactly successes in trials is given by the limit of a binomial distribution <br />(1) <br />Viewing the distribution as a function of the expected number of successes <br />(2) <br />instead of the sample size for fixed , equation (2) then becomes <br />(3) <br />Letting the sample size become large, the distribution then approaches <br />(4) (5) (6) (7) (8) <br />which is known as the Poisson distribution (Papoulis 1984, pp. 101 and 554; Pfeiffer and Schum 1973, p. 200). Note that the sample size has completely dropped out of the probability function, which has the same functional form for all values of . <br />The Poisson distribution is implemented in Mathematica as PoissonDistribution[mu]. <br />As expected, the Poisson distribution is normalized so that the sum of probabilities equals 1, since <br />(9) <br />The ratio of probabilities is given by <br />(10) <br />The Poisson distribution reaches a maximum when <br />(11) <br />where is the Euler-Mascheroni constant and is a harmonic number, leading to the transcendental equation <br />(12) <br />which cannot be solved exactly for . <br />The moment-generating function of the Poisson distribution is given by <br />(13) (14) (15) (16) (17) (18) <br />so <br />(19) (20) <br />(Papoulis 1984, p. 554). <br />The raw moments can also be computed directly by summation, which yields an unexpected connection with the exponential polynomial and Stirling numbers of the second kind, <br />(21) <br />known as Dobiński's formula. Therefore, <br />(22) (23) (24) <br />The central moments can then be computed as <br />(25) (26) (27) <br />so the mean, variance, skewness, and kurtosis are <br />(28) (29) (30) (31) (32) <br />The characteristic function for the Poisson distribution is <br />(33) <br />(Papoulis 1984, pp. 154 and 554), and the cumulant-generating function is <br />(34) <br />so <br />(35) <br />The mean deviation of the Poisson distribution is given by <br />(36) <br />The Poisson distribution can also be expressed in terms of <br />(37) <br />the rate of changes, so that <br />(38) <br />The moment-generating function of a Poisson distribution in two variables is given by <br />(39) <br />If the independent variables , , ..., have Poisson distributions with parameters , , ..., , then <br />(40) <br />has a Poisson distribution with parameter <br />(41) <br />This can be seen since the cumulant-generating function is <br />(42) (43) <br />A generalization of the Poisson distribution has been used by Saslaw (1989) to model the observed clustering of galaxies in the universe. The form of this distribution is given by <br />(44) <br />where is the number of galaxies in a volume , , is the average density of galaxies, and , with is the ratio of gravitational energy to the kinetic energy of peculiar motions, Letting gives <br />(45) <br />which is indeed a Poisson distribution with . Similarly, letting gives . <br />The Poisson Distribution was developed by the French mathematician Simeon Denis Poisson in 1837.<br />The Poisson random variable satisfies the following conditions:<br />On this page...<br />Mean and variance of Poisson distribution <br />The number of successes in two disjoint time intervals is independent.<br />The probability of a success during a small time interval is proportional to the entire length of the time interval.<br />Apart from disjoint time intervals, the Poisson random variable also applies to disjoint regions of space.<br />Applications<br />the number of deaths by horse kicking in the Prussian army (first application)<br />birth defects and genetic mutations<br />rare diseases (like Leukemia, but not AIDS because it is infectious and so not independent) - especially in legal cases<br />car accidents<br />traffic flow and ideal gap distance<br />number of typing errors on a page<br />hairs found in McDonald's hamburgers<br />spread of an endangered animal in Africa<br />failure of a machine in one month<br />The probability distribution of a Poisson random variable X representing the number of successes occurring in a given time interval or a specified region of space is given by the formula:<br />where<br />x = 0, 1, 2, 3...<br />e = 2.71828 (but use your calculator's e button)<br />μ = mean number of successes in the given time interval or region of space<br />Mean and Variance of Poisson Distribution<br />If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ.<br />E(X) = μ<br />and <br />V(X) = σ2 = μ<br />Note: In a Poisson distribution, only one parameter, μ is needed to determine the probability of an event. <br />EXAMPLE 1<br />A life insurance salesman sells on the average 3 life insurance policies per week. Use Poisson's law to calculate the probability that in a given week he will sell<br />(a) some policies <br />(b) 2 or more policies but less than 5 policies.<br />(c) Assuming that there are 5 working days per week, what is the probability that in a given day he will sell one policy?<br />Here, μ = 3<br />(a) " Some policies" means " 1 or more policies" . We can work this out by finding 1 minus the " zero policies" probability:<br />P(X > 0) = 1 − P(x0)<br />Now so <br />So <br />(b)<br />(c) Average number of policies sold per day: <br />So on a given day, <br />EXAMPLE 2<br />Twenty sheets of aluminum alloy were examined for surface flaws. The frequency of the number of sheets with a given number of flaws per sheet was as follows:<br />Number of flaws Frequency04132532445161<br />What is the probability of finding a sheet chosen at random which contains 3 or more surface flaws?<br />The total number of flaws is given by:<br />(0 × 4) + (1 × 3) + (2 × 5) + (3 × 2) + (4 × 4) + (5 × 1) + (6 × 1) = 46 <br />So the average number of flaws for the 20 sheets is given by:<br />The required probability is: <br />Histogram of Probabilities <br />We can see the predicted probabilities for each of " No flaws" , " 1 flaw" , " 2 flaws" , etc on this histogram.<br />[The histogram was obtained by graphing the following function for integer values of x only. <br />Then the horizontal axis was modified appropriately.] <br />EXAMPLE 3<br />If electricity power failures occur according to a Poisson distribution with an average of 3 failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week.<br />The average number of failures per week is: <br />" Not more than one failure" means we need to include the probabilities for " 0 failures" plus " 1 failure" . <br />EXAMPLE 4<br />Vehicles pass through a junction on a busy road at an average rate of 300 per hour.<br />Find the probability that none passes in a given minute.<br />What is the expected number passing in two minutes?<br />Find the probability that this expected number actually pass through in a given two-minute period.<br />The average number of cars per minute is: <br />(a) <br />(b) E(X) = 5 × 2 = 10 <br />(c) Now, with μ = 10, we have: <br />Histogram of Probabilities <br />Based on the function <br />we can plot a histogram of the probabilities for the number of cars per minute: <br />EXAMPLE 5<br />A company makes electric motors. The probability an electric motor is defective is 0.01. What is the probability that a sample of 300 electric motors will contain exactly 5 defective motors?<br />The average number of defectives in 300 motors is μ = 0.01 × 300 = 3<br />The probability of getting 5 defectives is:<br />NOTE: This problem looks similar to a binomial distribution problem, that we met in the last section.<br />If we do it using binomial, with n = 300, x = 5, p = 0.01 and q = 0.99, we get:<br />P(X = 5) = C(300,5)(0.01)5(0.99)295 = 0.10099 <br />We see that the result is very similar. We can use binomial distribution to approximate Poisson distribution (and vice-versa) under certain circumstances.<br />Histogram of Probabilities <br />

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