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(f) Two results related to giving alternative definitions of the Poisson process equivalent to the one we gave as Definition 21. (g) A result whose proof involves constructing a regular Markov chain from an irreducible Markov chain in such a way that the equilibrium distributions do not change.

(f) Two results related to giving alternative definitions of the Poisson process equivalent to the one we gave as Definition 21. (g) A result whose proof involves constructing a regular Markov chain from an irreducible Markov chain in such a way that the equilibrium distributions do not change.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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(f) Two results related to giving alternative definitions of the Poisson process equivalent to the one we gave as Definition 21. (g) A result whose proof involves constructing a regular Markov chain from an irreducible Markov chain in such a way that the equilibrium distributions do not change.
Transcribed Image Text:(f) Two results related to giving alternative definitions of the Poisson process equivalent to the one we gave as Definition 21. (g) A result whose proof involves constructing a regular Markov chain from an irreducible Markov chain in such a way that the equilibrium distributions do not change.
Definition 21. Let λ > 0. A continuous-time stochastic process (X(t): t≥ 0) is said to be a Poisson process of rate X if the following properties hold. (i) X (0) = 0. (ii) For any 0 <t₁ < t₂ < t3 <... < tn, the random variables X(t2) — X(t₁), X(t3) — X(t2), X(t4) — X(t3), …, X(tn) − X(tn_1) are mutually independent of one another¹¹. (iii) For any s, t ≥ 0, we have X (s+t) − X(s) ~ Po(λt).
Transcribed Image Text:Definition 21. Let λ > 0. A continuous-time stochastic process (X(t): t≥ 0) is said to be a Poisson process of rate X if the following properties hold. (i) X (0) = 0. (ii) For any 0 <t₁ < t₂ < t3 <... < tn, the random variables X(t2) — X(t₁), X(t3) — X(t2), X(t4) — X(t3), …, X(tn) − X(tn_1) are mutually independent of one another¹¹. (iii) For any s, t ≥ 0, we have X (s+t) − X(s) ~ Po(λt).
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