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I'm studying Morkov chains and I have a question about immigration process. Let's say I have two groups $X$ and $Y$ each individual of these groups give birth with the same rate $b$ and members of $X$ can leave their group and join to $Y$ with rate $l$. I define the transition with following table:

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I use birth death process for the transition probability for $X$ and get $X^{'}_{i}(t)=b(i-1)X_{i-1}(t)+l(i+1)X_{i+1}(t)-(b+l)iX_{i}(t)$. I think it is true but I am not sure about the second transition: $Y^{'}_{j}(t)=b(j-1)Y_{j-1}(t)+l(j-1)Y_{j-1}(t)-(b+l)jY_{j}(t)$ Could you please help me to find out my mistake?

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There was a very similar (if not identical) question very recently, but it seems to have been deleted. If that was yours, please explain. – joriki Nov 11 '12 at 11:06
@ joriki I think it was me:( I thought yesterday that I understand this process but now, I am confused again. – pcepkin Nov 11 '12 at 11:11
OK, that's not a problem -- just it might have been a good idea to explain it in the question, because it happens quite often that people delete and repose questions to get rid of critical comments or downvotes, which is an abuse of the system. Also you would have saved me the time to go looking for the duplicate :-) – joriki Nov 11 '12 at 11:28
@ joriki sorry, I will keep in mind for the next time. – pcepkin Nov 11 '12 at 11:38

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