(a) Use a detailed mathematical derivation to prove that the claim istrue.(b) Write an original problem using this concept and provide a correctsolution. Show that solving directly for P(N1(t) = k | N(t) = j) and transformingthe problem based on binomial random variables give the same answer. 1. Consider any Poisson Process with arrival rate A. Suppose that there are two typesof arrivals, where the first type of arrival comprise a proportion p of all arrivals and,likewise, the second type comprise a proportion 1 - p. Let N₁(t), N₂(t), and N(t)denote the number of type 1 arrivals, type 2 arrivals, and total arrivals, respectively,by time t.Claim: For any real number t ≥ 0 and non-negative integers n ≥ k ≥ 0, we haveP(N₁(t) = k|N(t) = n) = P(X = k),where X is a binomial random variable with success probability p and total number oftrials n.

Let S stand for the set of all conceivable type 1 and type 2 arrival sequences up to time t, and let…

(a) Use a detailed mathematical derivation to prove that the claim is true. (b) Write an original problem using this concept and provide a correct solution. Show that solving directly for P(N1(t) = k | N(t) = j) and transforming the problem based on binomial random variables give the same answer.

(a) Use a detailed mathematical derivation to prove that the claim is true. (b) Write an original problem using this concept and provide a correct solution. Show that solving directly for P(N1(t) = k | N(t) = j) and transforming the problem based on binomial random variables give the same answer.

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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