2. Let T exp(1) be exponentially distributed random time with intensity > 0. Consider a continuous-time stochastic process {X}2o defined by X₁ = (a) Write down the state space S of X₁. (b) Show for u < s t. P(X₁ = 1|X, = 0, Xµ = 0) = P(X₂ = 1|X, = 0), and conclude whether or not X, is a Markov process. (c) To verify that X, is time homogeneous, show for s < t that P(X₂ = 1|X, = 0) = P(X₁-s 1|Xo = 0). (d) The corresponding intensity matrix Q of X, is defined by 900 901 Q: =( :). = 910 911 qij = lim t→0 where the off-diagonal element q¡j, j ‡ i, i, j ¤ {0, 1}, is given by P(X₂ = j|Xo = i) t Derive the intensity matrix of the Markov process X. (e) Find the transition probability matrix P(t) of X.

2. Let T exp(1) be exponentially distributed random time with intensity > 0. Consider a continuous-time stochastic process {X}2o defined by X₁ = (a) Write down the state space S of X₁. (b) Show for u < s t. P(X₁ = 1|X, = 0, Xµ = 0) = P(X₂ = 1|X, = 0), and conclude whether or not X, is a Markov process. (c) To verify that X, is time homogeneous, show for s < t that P(X₂ = 1|X, = 0) = P(X₁-s 1|Xo = 0). (d) The corresponding intensity matrix Q of X, is defined by 900 901 Q: =( :). = 910 911 qij = lim t→0 where the off-diagonal element q¡j, j ‡ i, i, j ¤ {0, 1}, is given by P(X₂ = j|Xo = i) t Derive the intensity matrix of the Markov process X. (e) Find the transition probability matrix P(t) of X.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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