Question

In: Statistics and Probability

Small aircraft arrive at a private airport according to a given Poisson process at a rate...

Small aircraft arrive at a private airport according to a given Poisson process at a rate of 6 per hour.

  1. What is the probability that exactly 5 small aircraft arrive in the first hour (after opening) and exactly 7 arrive in the hour after that?
  2. What is the probability that fewer than 5 small aircraft arrive in the first hour and at least 10 arrive in the hour after that?
  3. What is the probability that exactly 5 small aircraft arrive in the first hour and exactly 12 aircraft arrive in the first two hours?

Solutions

Expert Solution

The arrival rate for the small aircrafts per hour here is given as 6 per hour. Therefore the arrival process here could be modelled as:

a) The probability that exactly 5 small aircraft arrive in the first hour (after opening) and exactly 7 arrive in the hour after that is computed here as:

P(X = 5) P(X = 7)

This is computed using poisson probability function here as:

Therefore 0.0221 is the required probability here.

b) The probability that fewer than 5 small aircraft arrive in the first hour and at least 10 arrive in the hour after that is computed here as:
P(X < 5)P(X >= 10) = P(X <= 4)*(1 - P(X <= 9))

This is computed in EXCEL here as:
=poisson.dist(4,6,TRUE)*(1-poisson.dist(9,6,TRUE))

0.0239 is the output here.

Therefore 0.0239 is the required probability here.

c) The probability that exactly 5 small aircraft arrive in the first hour and exactly 12 aircraft arrive in the first two hours

= The probability that exactly 5 small aircraft arrive in the first hour and exactly (12 - 5) = 7 aircraft arrive in the next hour

This is same as part a)

Therefore 0.0221 is the required probability here.


Related Solutions

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α...
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t. What is the probability that at least 7 small aircraft arrive during a 1-hour period? What is the probability that at least 11 small aircraft arrive during a 1-hour period? What is the probability that at least 23...
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α...
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t. (Round your answers to three decimal places.) (a) What is the probability that exactly 8 small aircraft arrive during a 1-hour period? What is the probability that at least 8 small aircraft arrive during a 1-hour period? What...
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α...
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t. (Round your answers to three decimal places.) (a) What is the probability that exactly 6 small aircraft arrive during a 1-hour period? What is the probability that at least 6 small aircraft arrive during a 1-hour period? What...
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α= 8 per hour
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α= 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter λ = 8t. (a) What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10? (b) What are the expected value and standard deviation of the number of small aircraft that arrive during...
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ=8t.
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ=8t. (Round your answers to three decimal places.)  (a) What is the probability that exactly 7 small aircraft arrive during a 1-hour period? What is the probability that at least 7 small aircraft arrive during a 1-hour period? What is the probability that at...
1.Suppose small aircraft arrive at a certain airport at a rate of 8 per hour. What...
1.Suppose small aircraft arrive at a certain airport at a rate of 8 per hour. What is the probability that at least 13 small aircrafts arrive during a given hour? 2.A particular telephone number is used to receive both voice calls and fax messages. Suppose that 30% of the incoming calls involve fax messages. Consider a sample of 20 incoming calls, what is the probability that at most 6 of the calls involve a fax message.
People arrive at a party according to a Poisson process of rate 30 per hour and...
People arrive at a party according to a Poisson process of rate 30 per hour and remain for an independent exponential time of mean 2 hours. Let X(t) be the number of people at the party at time t (in hours) after it started. Compute E[X(t)] and determine how long it takes to have on average more than 40 people at the party.
Emails arrive in an inbox according to a Poisson process with rate λ (so the number...
Emails arrive in an inbox according to a Poisson process with rate λ (so the number of emails in a time interval of length t is distributed as Pois(λt), and the numbers of emails arriving in disjoint time intervals are independent). Let X, Y, Z be the numbers of emails that arrive from 9 am to noon, noon to 6 pm, and 6 pm to midnight (respectively) on a certain day. (a) Find the joint PMF of X, Y, Z....
(20 pts) Inquiries arrive at an information center according to a Poisson process of rate 2...
(20 pts) Inquiries arrive at an information center according to a Poisson process of rate 2 inquiries per second. It takes a server an exponential amount of time with mean ½ second to answer each query. (a) (5 pts) If there are two servers. What is the probability that an arriving customer must wait to be served? (b) (5 pts) Find the mean number of customers in the system and the mean time spent in the system. (c) (10 pts)...
Customers arrive at a service facility according to a Poisson process of rate 5 /hour Let...
Customers arrive at a service facility according to a Poisson process of rate 5 /hour Let N(t) be the number of customers that have arrived up to time t hours). a. What is the probability that there is at least 2 customer walked in 30 mins? b.If there was no customer in the first30 minutes, what is the probability that you have to wait in total of more than 1 hours for the 1 st customer to show up? c.For...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT