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)

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...

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