Pure Birth Process












4












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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.
    $endgroup$
    – 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
















4












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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.
    $endgroup$
    – 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














4












4








4





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited May 6 '13 at 20:24







acwang123

















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


















  • $begingroup$
    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










1 Answer
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$begingroup$

See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)






share|cite|improve this answer









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    active

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    0












    $begingroup$

    See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)






        share|cite|improve this answer









        $endgroup$



        See these notes, pages 233-234 work through a very similar problem. (Rather than being $klambda$ your rate is $(N-k)lambda$.)







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered May 30 '13 at 13:34









        GarethGareth

        947512




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