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
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Transcribed Image Text: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 that
=
1, T ≤t,
0,
T>t.
P(X₁ = 1|X, = 0, X₁ = 0) = P(X₂ = 1|X, = 0),
Xµ
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(X-s = 1|Xo = 0).
(d) The corresponding intensity matrix Q of X, is defined by
e=(
qij
2010).
900 901
910 911
where the off-diagonal element qij, j ‡ i, i, j € {0, 1}, is given by
P(X₁ = j|Xo = i)
t
= lim
t→0
Derive the intensity matrix of the Markov process X.
(e) Find the transition probability matrix P(t) of X.
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