For a general birth-and-death process with state space S = {0, 1, 2, ...}, birth rates A; (i e S) and death rates u; (i = 1,2, 3, ...), let Y; denote the time, from state i e S, for the process to hit state i + 1. (a) Determine E[Yo] and show that 1 + di E[Y;] E[Y;-1] for i = 1,2, 3,... Hint: Condition on whether the first transition is a birth or a death. (b) Compute E[Y;] explicitly (i.e. in closed-form) for the case A; = X and l; = u for all i.
For a general birth-and-death process with state space S = {0, 1, 2, ...}, birth rates A; (i e S) and death rates u; (i = 1,2, 3, ...), let Y; denote the time, from state i e S, for the process to hit state i + 1. (a) Determine E[Yo] and show that 1 + di E[Y;] E[Y;-1] for i = 1,2, 3,... Hint: Condition on whether the first transition is a birth or a death. (b) Compute E[Y;] explicitly (i.e. in closed-form) for the case A; = X and l; = u for all i.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![For a general birth-and-death process with state space S = {0, 1, 2, ...},
birth rates A; (i e S) and death rates u; (i = 1,2, 3, ...), let Y; denote the time, from state
i e S, for the process to hit state i + 1.
(a) Determine E[Yo] and show that
1
+
di
E[Y;]
E[Y;-1]
for i = 1,2, 3,...
Hint: Condition on whether the first transition is a birth or a death.
(b) Compute E[Y;] explicitly (i.e. in closed-form) for the case A; = X and l; = u for all i.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9022cbcf-8dfd-480f-9eac-ecde7b5a2238%2Fb03163e7-be49-42b9-a137-d3a51d257164%2Ftwe2z6g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For a general birth-and-death process with state space S = {0, 1, 2, ...},
birth rates A; (i e S) and death rates u; (i = 1,2, 3, ...), let Y; denote the time, from state
i e S, for the process to hit state i + 1.
(a) Determine E[Yo] and show that
1
+
di
E[Y;]
E[Y;-1]
for i = 1,2, 3,...
Hint: Condition on whether the first transition is a birth or a death.
(b) Compute E[Y;] explicitly (i.e. in closed-form) for the case A; = X and l; = u for all i.
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