【正文】
er (ppois is a cumulative distribution). ppois(2,) [1] Consider a collection of graphs for different values of ? ?=3 ?=4 ?=5 ?=6 ?=10 In the last case, the probability of 20 arrivals is no longer negligible, so values up to, say, 30 would have to be considered. Properties of Poisson The mean and variance are both equal to ?. The sum of independent Poisson variables is a further Poisson variable with mean equal to the sum of the individual means. As well as cropping up in the situations already mentioned, the Poisson distribution provides an approximation for the Binomial distribution. Approximation: If n is large and p is small, then the Binomial distribution with parameters n and p, ( B(n。Example A student attempts a multiple choice exam (options A to F for each question), but having done no work, selects his answers to each question by rolling a fair die (A = 1, B = 2, etc.). If the exam contains 100 questions, what is the probability of obtaining a mark below 20? Simulation Now, let us simulate a large number of realisations of students using this random method of answering multiple choice questions. We still require the same Binomial distribution with n=100 and a= This can be done on R using the mand rbinom. 16 For example, let’s simulate 1000 students. xsim=rbinom(1000,100,1/6) xsim [1] 18 22 9 17 18 20 21 16 8 18 11 16 16 13 16 14 25 15 16 17 [21] 13 25 11 24 17 16 13 21 10 17 18 10 17 18 19 17 19 15 13 12 [41] 15 11 21 23 19 14 19 25 23 19 20 17 17 15 16 14 13 16 17 14 [61] 24 21 19 8 18 20 22 16 15 20 19 17 13 15 13 21 22 12 12 1