(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...
Related questions
Question
(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.
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.

Transcribed Image Text:1. Consider any Poisson Process with arrival rate A. Suppose that there are two types
of 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 have
P(N₁(t) = k|N(t) = n) = P(X = k),
where X is a binomial random variable with success probability p and total number of
trials n.
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