AS/A2 Stats 1 Stats 3

S2 Topic 1: Binomial and Poisson distributions
Poisson probabilites 2
The Poisson distribution is usually employed for modeling systems where the probability of an event occurring is very low, but the number of opportunities for such occurrence is very high, for example radioactive decay or rare disease clusters. Choose a value for l from 1 to 30 and the number of trials, N from 100 up to 1,000,000,000, then press Compute. The Poisson probabilites, P(X=x), will be evaluated for values of x from 0 to 59. The cumulative probability P(X£x) will also be given. The last two columns show the expected number of trials, E, that will have x occurrances and the cumulative expected number, CE. Here are some problems. Assume that on average, each square mile of the earth's surface is struck by one meteor of a certain size each year and that the total square miles of the earth's surface is 200,000,000. How many square miles of the earth's surface will be hit by no meteors at all, and how many will be hit by more than 5? On average in a week you are late to school or work once. How many weeks will you be late at least once in a year? On average a box of 250 apples contains 15 bad ones. What is the proability of getting more than 2 bad apples in a box? A small car hire firm owns five cars. On average the weekday demand is for 2 cars and the demand is 3 during the weekend. What is the probability of refusing a customer on (a) Monday, (b) Saturday, (c) during the weekend?

l 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 2 2.5 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30	N 7 10 12 50 52 100 200 250 500 1,000 2,000 5,000 10,000 100,000 1,000,000 10,000,000 50,000,000 100,000,000 200,000,000 250,000,000 500,000,000 1,000,000,000

x	P(X=x)	P(X£x)	E	CE

Answers 73,575,888 square miles will escape bombardment totally, while 731,969 square miles will be hit five or more times. (200,000,000 - 199,268,031, the number of square miles hit 4 or fewer times.) You will be late 33 (52-19) weeks in the year. The probability of more than two apples = 0.1912 (=1-0.8808) (a) 0.0166 (b) 0.0839, (c) 0.1608 A note about graphic calculators A graphic calculator, for example the TI-83 or similar model, can work out Poisson probabilities for you. Suppose that X ~ P(5): P(X=2) is given by poissonpdf(5,2)=.0842243375 P(X £ 2) is given by poissoncdf(5,2)=.1246520195 © MathsNet 2001