# expected time to enter a state in birth-death process

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.

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?

thanks

• $I_i=1$ doesn't mean that the next event is a birth. $I$ stands for the direction of the first transition from that state. – gunes Dec 5 '18 at 14:43
• 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 $\frac{1}{\lambda_i + \mu_i}$ – john_w Dec 7 '18 at 6:59