I have a question regrading the expected time of entering a state in a birth-death process. Specifically I don't quite understand Page 378 Equation 6.3 of the book here.

It is about birth-death process.

In it, it says if the birth rate is $\lambda_i$, and death rate is $\mu_i$, if we define $T_i$ to be the time from state $i$ to state $i+1$ ( in this case the state i means population is of size i at time T). Then it says if we condition on whether the first transition from i is to i-1, or i+1, we could calculate the expected value of $T_i$.

Define: $I_i = 1$ if the first transition out of i is to i+1. And $I_i = 0$ if the first transition out of i is to i-1.

Then it says $E(T_i|I_i=1) = \frac{1}{\lambda_i + \mu_i}$.

I thought the expected Time to ANY event (in this case, either death or birth) would be $\frac{1}{\lambda_i + \mu_i}$ and expected time to birth be $\frac{1}{\lambda_i}$.

So if given $I_i = 1$, shouldn't expected time to the next state be the expected time to the birth event? i.e. shouldn't $E(T_i|I_i=1) = \frac{1}{\lambda_i}$?

But professor sheldon ross (the author) is saying it is $E(T_i|I_i=1) = \frac{1}{\lambda_i + \mu_i}$. Could someone kindly explain about my misunderstanding?


  • $\begingroup$ $I_i=1$ doesn't mean that the next event is a birth. $I$ stands for the direction of the first transition from that state. $\endgroup$ – gunes Dec 5 '18 at 14:43
  • $\begingroup$ Hello gunes, but could you explain more about what you means by "the direction of the first transition"? and how the $E(T_i | I_i=1) $ arrives at $\frac{1}{\lambda_i + \mu_i}$ $\endgroup$ – john_w Dec 7 '18 at 6:59

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