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X is a simple birth death process with birth rate $\lambda n$ and death rate $\mu n$ Embedded within a simple birth death process is a simple random walk. Let $Y_n $ be the value of X at the time of the nth event. $Y_0=X_0=1 $

Then

$Y_{n+1} = Y_n + 1 $ with probability $\lambda / (\lambda + \mu)$

And

$Y_{n+1} = Y_n-1 $ with probability $\mu / (\lambda + \mu)$

Provided $Y_n > 0$

Since X doesn't explode, extinction occurs in finite time iff $Y_n =0, n<\infty$

So $P[Y_n =0, n<\infty]. = 1$ if $\mu\geq\lambda$ Or $(\mu/\lambda)^X_0 $ if $\mu<\lambda$

where we use our analysis of discrete-time simple random walks,

How is the analysis of discrete simple random walks used here, I don't understand where these probabilities come from, how are they worked out? Thanks

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