# Birth & Death process - Combining Transition rates

I think I'm missing a fundamental step in regards to how to combine two exponential distributions in the context of this problem.

If we have a birth and death process where birth rate ~ $exp(\lambda_n)$ and the death rate ~ $exp( \mu_n )$, and state can go from n to $n-1$ or to $n+1$, why is it that $v_i=\lambda_i+\mu_i$?

Note that the transition probability(time spent in state i before transition) is $T_i$ ~ $exp(v_i)$ and this is where $v_i$ is used

• Is this from a course or textbook? If so, please add the [self-study] tag & read its wiki. – gung Apr 4 '15 at 22:22
• It is but I only use it as an example to understand a particular step of the provided solution. Should I still mark it as such – Borat.sagdiyev Apr 4 '15 at 22:29
• Probably, & read the wiki. We will give you hints & help you think through the problem for yourself, but probably not just give you a full answer. – gung Apr 4 '15 at 22:31
• it is a very short question. Presumably answerable in 1 sentence. It's a property of exponential variables that I'm just not seeing. Please read what I wrote above about the solution being provided and myself just inquiring about one of the steps – Borat.sagdiyev Apr 4 '15 at 22:38
• I've edited the question to make it clear that what I'm asking about is a fundamental property of birth and death processes and not a textbook problem – Borat.sagdiyev Apr 4 '15 at 22:48

I found the property I was looking for. IT applies to this problem because in order to change state, either a birth or a death must occur, whichever takes less time.

If $X_1 ... X_n$ are i.i.d exponential R.V's with parameters $\lambda_1...\lambda_n$ then,

$min_i$ $X_i$ is an exponential R.V. with parameter $\lambda_1+...+\lambda_n$