Homework Help is Here – Start Your Trial Now!
Math
Probability
8. Let (N₁ (t), t≥ 0} (type 1 arrivals) and {N₂(t), t≥ 0} (type 2 arrivals) be two indepen- dent Poisson processes with rates A₁ = 1 and X₂ =2, respectively. Let {N(t), t > 0} be the merged process, where N(t) = N₁(t) + N₂(t) for all t≥ 0. (a) Find the probability that N(1) = 2 and N(2) = 5. (b) Given that N(1) = 2 of the merged process, find the probability that №₁(1) = 1. Hints: Recall advanced probability examples. 2 (c) Find the probability that 3 type-1 arrivals occur during (0.5, 2] and 2 type-2 arrivals occur during (1.5,2]. (d) A customer just arrived. What is the probability it is of type 1? (e) Find the probability that the second arrival in N₁ (t) occurs before the third arrival in N₂(t). Hints: One way to solve this problem (=part (e)) is to think of N₁ (t) and N₂(t) as two processes obtained from splitting a Poisson process. Then, turn this problem into the following: Given that there is a total of 4 arrivals in the merged process, what needs to happen for the second arrival of type 1 to occur before the third arrival of type 2. Obtain an answer for this conditional probability.

8. Let (N₁ (t), t≥ 0} (type 1 arrivals) and {N₂(t), t≥ 0} (type 2 arrivals) be two indepen- dent Poisson processes with rates A₁ = 1 and X₂ =2, respectively. Let {N(t), t > 0} be the merged process, where N(t) = N₁(t) + N₂(t) for all t≥ 0. (a) Find the probability that N(1) = 2 and N(2) = 5. (b) Given that N(1) = 2 of the merged process, find the probability that №₁(1) = 1. Hints: Recall advanced probability examples. 2 (c) Find the probability that 3 type-1 arrivals occur during (0.5, 2] and 2 type-2 arrivals occur during (1.5,2]. (d) A customer just arrived. What is the probability it is of type 1? (e) Find the probability that the second arrival in N₁ (t) occurs before the third arrival in N₂(t). Hints: One way to solve this problem (=part (e)) is to think of N₁ (t) and N₂(t) as two processes obtained from splitting a Poisson process. Then, turn this problem into the following: Given that there is a total of 4 arrivals in the merged process, what needs to happen for the second arrival of type 1 to occur before the third arrival of type 2. Obtain an answer for this conditional probability.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
See similar textbooks
icon
Related questions
Question
VERY URGENT
8. Let (N₁ (t), t2 0} (type 1 arrivals) and {N₂(t), t≥ 0} (type 2 arrivals) be two indepen- dent Poisson processes with rates A₁ = 1 and A₂ = 2, respectively. Let {N(t), t > 0} be the merged process, where N(t) = N₁(t) + N₂(t) for all t≥ 0. (a) Find the probability that N(1) = 2 and N(2) = 5. (b) Given that N(1)=2 of the merged process, find the probability that N₁ (1) = 1. Hints: Recall advanced probability examples. 2 (c) Find the probability that 3 type-1 arrivals occur during (0.5, 2] and 2 type-2 arrivals occur during (1.5,2]. (d) A customer just arrived. What is the probability it is of type 1? (e) Find the probability that the second arrival in N₁ (t) occurs before the third arrival in N₂(t). Hints: One way to solve this problem (=part (e)) is to think of N₁ (t) and N₂(t) as two processes obtained from splitting a Poisson process. Then, turn this problem into the following: Given that there is a total of 4 arrivals in the merged process, what needs to happen for the second arrival of type 1 to occur before the third arrival of type 2. Obtain an answer for this conditional probability.
Transcribed Image Text:8. Let (N₁ (t), t2 0} (type 1 arrivals) and {N₂(t), t≥ 0} (type 2 arrivals) be two indepen- dent Poisson processes with rates A₁ = 1 and A₂ = 2, respectively. Let {N(t), t > 0} be the merged process, where N(t) = N₁(t) + N₂(t) for all t≥ 0. (a) Find the probability that N(1) = 2 and N(2) = 5. (b) Given that N(1)=2 of the merged process, find the probability that N₁ (1) = 1. Hints: Recall advanced probability examples. 2 (c) Find the probability that 3 type-1 arrivals occur during (0.5, 2] and 2 type-2 arrivals occur during (1.5,2]. (d) A customer just arrived. What is the probability it is of type 1? (e) Find the probability that the second arrival in N₁ (t) occurs before the third arrival in N₂(t). Hints: One way to solve this problem (=part (e)) is to think of N₁ (t) and N₂(t) as two processes obtained from splitting a Poisson process. Then, turn this problem into the following: Given that there is a total of 4 arrivals in the merged process, what needs to happen for the second arrival of type 1 to occur before the third arrival of type 2. Obtain an answer for this conditional probability.
9. Alex goes to Tim Horton's to get coffee, on average, 3 times a week according to a Poisson Process. Now, Tim Horton's is running a promotion. Every coffee cup has a sticker. With probability 0.50, the sticker will not yield any prizes. With probability 0.30, one gets a free cup of coffee worth $2. With probability 0.15, one can win a free meal worth $10. With 0.05 probability, a sticker will yield a grand prize of a car worth $30,000. What is Alex's expected earnings in the month of February?
Transcribed Image Text:9. Alex goes to Tim Horton's to get coffee, on average, 3 times a week according to a Poisson Process. Now, Tim Horton's is running a promotion. Every coffee cup has a sticker. With probability 0.50, the sticker will not yield any prizes. With probability 0.30, one gets a free cup of coffee worth $2. With probability 0.15, one can win a free meal worth $10. With 0.05 probability, a sticker will yield a grand prize of a car worth $30,000. What is Alex's expected earnings in the month of February?
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 30 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer