Talk:Birth process

Typo?

The forward equation seems to have a typo. lamda_i should be lambda_j? If this is a typo, this in turn leads to a problem in the p_ij(t) solution that follows? Soungliew (talk) 03:51, 19 October 2023 (UTC)

@Soungliew: this is testing the limits of my memory and Markov chain knowledge. Unfortunately I no longer have access to the source I used (Ross, 2010, p. 389). I presume you are talking about:

$p'_{i,j}(t)=\lambda _{j-1}p_{i,j-1}(t)-\lambda _{i}p_{i,j}(t)$

I have changed the $\lambda _{i}$ to $\lambda _{j}$ after consulting some notes online about birth and death processes and substituting the death rate $\mu _{x}=0$, which matches what you are saying. However, since I took the equations from the source, unless I made multiple typos or the textbook made an error, the solutions $p_{i,j}(t)$ should still be correct.

Do you think they contain a separate error and if so what is it? — Bilorv (talk) 21:34, 24 October 2023 (UTC)

I looked at the revision. It seems that the backward equation now has a typo (the one referencing [6]).

For backward equations, it is the initial state that varies while keeping the final state constant. So, on the RHS, I believe p_i,j+1 should be p_i+1,j.

The forward equation looks okay now (the one referencing [7]).

However, the equation below that with the integration from 0 to t may have a typo. lambda_i in exp(-lambda_i t) should be lambda_j. Direct verification by substituting into the forward differential equation indicates that this is a typo. Soungliew (talk) 23:50, 24 October 2023 (UTC)

@Soungliew: I'm not quite sure how these errors came in but I've looked up the backwards equations in other sources and differentiated the given $p_{i,j}(t)$ and agree in both cases. Thanks for pointing these out—in future feel free to make corrections to the article directly and add a talk page comment if the edit summary field isn't big enough to contain the reasoning. — Bilorv (talk) 15:49, 26 October 2023 (UTC)