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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!

acwang123
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  • 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. – A.P. May 06 '13 at 20:07
  • I updated the question to follow your suggested guidelines. Thanks! – acwang123 May 06 '13 at 20:12
  • I've at least found the definition of pure-birth process http://en.wikipedia.org/wiki/Birth-death_process#Examples – Brady Trainor May 06 '13 at 20:17
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    Can you construct the detailed balance equation from this? – Alex May 06 '13 at 20:19
  • 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! – acwang123 May 06 '13 at 20:24
  • Or take the derivative of the solution, that might help you see the connection to the difference equation. Can you type the difference equation in your question? – Brady Trainor May 06 '13 at 20:28

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