Question

the statement of the problem:

"Buses arrive at a bus station at a rate 2 buses per minute. the arrival of buses occurs at random times according to a Poisson scatter on (0, infinity). Show all your work for each part."

a. what is the probability that no bus arrives between t = 0 and t = 30 seconds?

b. what is the probability that the third bus arrives after t = 3 minutes?

c. what is the probability that the 5-th bus arrives within 1 minute of the 4-th bus?

d. what is the probability that the 1-st bus arrives in less than 1 minute and fewer than 2 buses arrive between t = 3 and t = 5 minutes.

Answer 1

a) The average number of bus arrivals is 2 per minute. Therefore per 30 seconds, average rate of arrival would be given as: 2/2 = 1 bus per 30sec. The probability that no bus arrives between t = 0 and t = 30 sec is computed here as:

= e-1 = 0.3679

Therefore 0.3679 is the required probability here.

b) Now the probability that the third bus arrives after t = 3 minutes is computed here as:
= Probability of < 3 arrivals in first 3 minutes

For 3 arrivals the average number of arrivals is given as: 3*2 = 6. Therefore the probability here is computed as:

Therefore 0.0620 is the required probability here.

c) Now the probability that the 5-th bus arrives within 1 minute of the 4-th bus is computed here as:

= Probability that the 5th bus arrives in less than 1 minute.

= 1 - Probability of no arrival in next 1 minute

= 1 - e-2

= 0.8647

Therefore 0.8647 is the required probability here.

d) Now the probability that the 1-st bus arrives in less than 1 minute and fewer than 2 buses arrive between t = 3 and t = 5 minutes.

= Probability there is at least one arrival in 1st minute * Probability that there are 0 or 1 arrival from 3 to 5 minutes.

Therefore 0.0792 is the required probability here.