Question

7.1. Cars arrive to a toll booth 24 hours per day according to a Poisson process with a mean rate of 15 per hour. (a) What is the expected number of cars that will arrive to the booth between 1:00 p.m. and 1:30 p.m.? (b) What is the expected length of time between two consecutively arriving cars! (c) It is now 1:12 p.m. and a car has just arrived. What is the expected number of cars that will arrive between now and 1:30 p.m.? (d) It is now 1:12 p.m. and a car has just arrived. What is the probability that two more cars will arrive between now and 1:30 p.m.? (e) It is now 1:12 p.m. and the last car to arrive came at 1:05 p.m. What is the probability that no additional cars will arrive before 1:30 p.m.? f) It is now 1:12 p.m. and the last car to arrive came at 1:05 p.m. What is the expected length of time between the last car to arrive and the next car to arrive?

Answer 1

a)

mean rate=15 car per hour

expected number of cars between 1:00 pm to 1:30 pm i.e during half hour duration = 15/2 = 7.5 cars

b)expected length of time betwwen two consective arriving cars= 1 / mean = 1/15 = 0.067 hour

c)

expected number of cars between 1:12 pm to 1:30 pm i.e during 18 minutes interval = 15/60 * 18 = 4.5 cars

d)

poisson probability distribution

P(X=x) = e-λλx/x!

here Mean/Expected number of events of interest: λ =                4.5 cars

P ( X =    2   ) = e ^   -4.5   *    4.5   ^    2   /   2   ! =0.1125(answer)

e)

expected number of cars between 1:05 pm to 1:30 pm i.e during 25 minutes interval = 15/60 * 25 = 6.25 cars

poisson probability distribution

P(X=x) = e-λλx/x!

P ( X =    0   ) = e ^(-6.5)* 6.5 ^0/   0! = 0.0019(answer)

f)

expected length = 1/6.25 = 0.16 hours or 9.6minutes