Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao a > 0, A₁ = 0, μo = 0, and μ₁ => 0. (a) Show that the infinitesimal generator matrix A = Thus, for n = 1,2,3,..., A" = (-a - b)-1A holds. -a a b satisfies the property A² = (a + b)A. (b) Using the result from part (a) to compute the matrix exponential etA in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim∞ P(t) and use it to find the stationary distribution.
Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao a > 0, A₁ = 0, μo = 0, and μ₁ => 0. (a) Show that the infinitesimal generator matrix A = Thus, for n = 1,2,3,..., A" = (-a - b)-1A holds. -a a b satisfies the property A² = (a + b)A. (b) Using the result from part (a) to compute the matrix exponential etA in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim∞ P(t) and use it to find the stationary distribution.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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Step 1: Write the given information.
VIEWStep 2: Show that the infinitesimal generator matrix A satisfies the given property.
VIEWStep 3: Compute the matrix exponential e^(tA) in closed form.
VIEWStep 4: Determine the transition probability matrix P(t).
VIEWStep 5: Compute the stationary distribution using using P(t).
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