Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter ? =20 (suggested in the article “Dynamic Ride Sharing: Theory and Practice,” J. of Transp. Engr., 1997: 308–312). What is the probability that the number of drivers will a.? e at most 10? b.?? xceed 20? c.? e between 10 and 20, inclusive? Be strictly between 10 and 20? d.? ?Be within 2 standard deviations of the mean value?

Solution Step 1 : Here X is the number of drivers who traveled between a particular origin and destination during a designated time period follows a Poisson distribution with parameter =20 . we have to find probabilities corresponds to different X values. Step 2 : a) We have to find the probability that the number of drivers will be at most 10. It is given that the random variable X follows poisson distribution with parameter (mean) = 20. The probability mass function of poisson distribution is x e P(x) = x! , x= 0,1,2,3,... So the probability that the number of drivers will be at most 10 is 10 e x P(X10)= x! x=0 10 e20 20x = x! x=0 = P(x=0)+p(x=1)+p(x=2)+...+P(x=10) = 0.00+0.000+0.00+0.00+0.00001+.00005+0.00018+...+.00581 = 0.0108 We can find this probability in excel also by using the function (POISSON(X,MEAN,CUMULATIVE= FALSE),then SUM(CELL NUMBERS) ).