Pure Birth Process
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I encountered this problem while trying out various practice problems to study for my stochastic processes test. (It's not homework, it's just a practice problem).
Consider a pure birth process on the states 0,1,...,n for which $lambda_k = (N-k)lambda$ for $k = 0,1,...,N$. Suppose that $X(0) = 0$. Determine $P_n(t) = Pr{X(t)=n}$ for $n = 0,1,2$
The solution that I was given was $P_n(t) = {N choose n}e^{-(N-n)lambda t}(1-e^{-lambda t})^n$.
Could someone please explain this problem's solution to me? I can't seem to derive it from any formula that I have been given by the class. Any help would be greatly appreciated!
stochastic-processes
asked May 6 '13 at 19:46
acwang123acwang123
213
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|
show 1 more comment
$begingroup$
I encountered this problem while trying out various practice problems to study for my stochastic processes test. (It's not homework, it's just a practice problem).
Consider a pure birth process on the states 0,1,...,n for which $lambda_k = (N-k)lambda$ for $k = 0,1,...,N$. Suppose that $X(0) = 0$. Determine $P_n(t) = Pr{X(t)=n}$ for $n = 0,1,2$
The solution that I was given was $P_n(t) = {N choose n}e^{-(N-n)lambda t}(1-e^{-lambda t})^n$.
Could someone please explain this problem's solution to me? I can't seem to derive it from any formula that I have been given by the class. Any help would be greatly appreciated!
stochastic-processes
asked May 6 '13 at 19:46
acwang123acwang123
213
$endgroup$
$begingroup$
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.
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– A.P.
May 6 '13 at 20:07
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I updated the question to follow your suggested guidelines. Thanks!
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– acwang123
May 6 '13 at 20:12
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I've at least found the definition of pure-birth process en.wikipedia.org/wiki/Birth-death_process#Examples
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– Brady Trainor
May 6 '13 at 20:17
1
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Can you construct the detailed balance equation from this?
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– Alex
May 6 '13 at 20:19
$begingroup$
I've updated the question with the solution. However, the solution doesn't make sense to me, and is hence what I'm asking for help on. Thanks guys!
$endgroup$
– acwang123
May 6 '13 at 20:24
|
show 1 more comment
$begingroup$
I encountered this problem while trying out various practice problems to study for my stochastic processes test. (It's not homework, it's just a practice problem).
Consider a pure birth process on the states 0,1,...,n for which $lambda_k = (N-k)lambda$ for $k = 0,1,...,N$. Suppose that $X(0) = 0$. Determine $P_n(t) = Pr{X(t)=n}$ for $n = 0,1,2$
The solution that I was given was $P_n(t) = {N choose n}e^{-(N-n)lambda t}(1-e^{-lambda t})^n$.
Could someone please explain this problem's solution to me? I can't seem to derive it from any formula that I have been given by the class. Any help would be greatly appreciated!
stochastic-processes
asked May 6 '13 at 19:46
acwang123acwang123
213
$endgroup$
I encountered this problem while trying out various practice problems to study for my stochastic processes test. (It's not homework, it's just a practice problem).
Consider a pure birth process on the states 0,1,...,n for which $lambda_k = (N-k)lambda$ for $k = 0,1,...,N$. Suppose that $X(0) = 0$. Determine $P_n(t) = Pr{X(t)=n}$ for $n = 0,1,2$
The solution that I was given was $P_n(t) = {N choose n}e^{-(N-n)lambda t}(1-e^{-lambda t})^n$.
Could someone please explain this problem's solution to me? I can't seem to derive it from any formula that I have been given by the class. Any help would be greatly appreciated!
stochastic-processes
stochastic-processes
asked May 6 '13 at 19:46
acwang123acwang123
213
asked May 6 '13 at 19:46
acwang123acwang123
213
edited May 6 '13 at 20:24
acwang123
asked May 6 '13 at 19:46
acwang123acwang123
213
asked May 6 '13 at 19:46
acwang123acwang123
213
asked May 6 '13 at 19:46
acwang123acwang123
213
213
$begingroup$
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.
$endgroup$
– A.P.
May 6 '13 at 20:07
$begingroup$
I updated the question to follow your suggested guidelines. Thanks!
$endgroup$
– acwang123
May 6 '13 at 20:12
$begingroup$
I've at least found the definition of pure-birth process en.wikipedia.org/wiki/Birth-death_process#Examples
$endgroup$
– Brady Trainor
May 6 '13 at 20:17
1
$begingroup$
Can you construct the detailed balance equation from this?
$endgroup$
– Alex
May 6 '13 at 20:19
$begingroup$
I've updated the question with the solution. However, the solution doesn't make sense to me, and is hence what I'm asking for help on. Thanks guys!
$endgroup$
– acwang123
May 6 '13 at 20:24
|
show 1 more comment
$begingroup$
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.
$endgroup$
– A.P.
May 6 '13 at 20:07
$begingroup$
I updated the question to follow your suggested guidelines. Thanks!
$endgroup$
– acwang123
May 6 '13 at 20:12
$begingroup$
I've at least found the definition of pure-birth process en.wikipedia.org/wiki/Birth-death_process#Examples
$endgroup$
– Brady Trainor
May 6 '13 at 20:17
1
$begingroup$
Can you construct the detailed balance equation from this?
$endgroup$
– Alex
May 6 '13 at 20:19
$begingroup$
I've updated the question with the solution. However, the solution doesn't make sense to me, and is hence what I'm asking for help on. Thanks guys!
$endgroup$
– acwang123
May 6 '13 at 20:24
$begingroup$
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.
$endgroup$
– A.P.
May 6 '13 at 20:07
$begingroup$
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.
$endgroup$
– A.P.
May 6 '13 at 20:07
$begingroup$
I updated the question to follow your suggested guidelines. Thanks!
$endgroup$
– acwang123
May 6 '13 at 20:12
$begingroup$
I updated the question to follow your suggested guidelines. Thanks!
$endgroup$
– acwang123
May 6 '13 at 20:12
$begingroup$
I've at least found the definition of pure-birth process en.wikipedia.org/wiki/Birth-death_process#Examples
$endgroup$
– Brady Trainor
May 6 '13 at 20:17
$begingroup$
I've at least found the definition of pure-birth process en.wikipedia.org/wiki/Birth-death_process#Examples
$endgroup$
– Brady Trainor
May 6 '13 at 20:17
1
1
$begingroup$
Can you construct the detailed balance equation from this?
$endgroup$
– Alex
May 6 '13 at 20:19
$begingroup$
Can you construct the detailed balance equation from this?
$endgroup$
– Alex
May 6 '13 at 20:19
$begingroup$
I've updated the question with the solution. However, the solution doesn't make sense to me, and is hence what I'm asking for help on. Thanks guys!
$endgroup$
– acwang123
May 6 '13 at 20:24
$begingroup$
I've updated the question with the solution. However, the solution doesn't make sense to me, and is hence what I'm asking for help on. Thanks guys!
$endgroup$
– acwang123
May 6 '13 at 20:24
|
show 1 more comment
1 Answer
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See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)
answered May 30 '13 at 13:34
GarethGareth
947512
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add a comment |
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$begingroup$
See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)
answered May 30 '13 at 13:34
GarethGareth
947512
$endgroup$
add a comment |
$begingroup$
See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)
answered May 30 '13 at 13:34
GarethGareth
947512
$endgroup$
add a comment |
$begingroup$
See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)
answered May 30 '13 at 13:34
GarethGareth
947512
$endgroup$
See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)
answered May 30 '13 at 13:34
GarethGareth
947512
answered May 30 '13 at 13:34
GarethGareth
947512
answered May 30 '13 at 13:34
GarethGareth
947512
answered May 30 '13 at 13:34
GarethGareth
947512
947512
add a comment |
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$begingroup$
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level.
$endgroup$
– A.P.
May 6 '13 at 20:07
$begingroup$
I updated the question to follow your suggested guidelines. Thanks!
$endgroup$
– acwang123
May 6 '13 at 20:12
$begingroup$
I've at least found the definition of pure-birth process en.wikipedia.org/wiki/Birth-death_process#Examples
$endgroup$
– Brady Trainor
May 6 '13 at 20:17
1
$begingroup$
Can you construct the detailed balance equation from this?
$endgroup$
– Alex
May 6 '13 at 20:19
$begingroup$
I've updated the question with the solution. However, the solution doesn't make sense to me, and is hence what I'm asking for help on. Thanks guys!
$endgroup$
– acwang123
May 6 '13 at 20:24