1 25 Events occur according to a Poisson process with rate l

1. (25%) Events occur according to a Poisson process with rate lambda = 2 per hour. (a) What is the probability that more than 1 but fewer than 5 (i.e., 2, 3 or 4) events occur between the times of 8 A.M. and 10 A.M.? (b) What is the expected value of the number of events that occur between the times of S A.M. and 10 A.M.? (c) If we start counting events at 8 A.M., what time of day is the 5th event expected? (d) if we know the 10th event occurred at 3 P.M., what is the probability that the 11th event will occur before 4 P.M.? (e) if we know the 10th event occurred at 3 P.M., what is the probability that the 12th event will occur after 4 P.M.?

Solution

Px = c^x e^-c/x!

c=2*hours

Pmore than 1 = 4²e^-4/2! + 4³e^-4/3! + 4⁴e^-4/4! = 0.537

expected value = 2*t =4

5 th event is expected to occur at 5/2=2. hours latter that is 10 :30 AM

P11=2¹ e^-2/1!=0.27

P12 = 2²e^-2/2! =0.27