Consider a birth and death process $X(t)$ with birthrate $n(\lambda_1+\lambda_2)$ and death rate $n\mu$ for $n$ individuals at that time. We assume, that $\lambda_1$ and $\lambda_2$ are two independent ways of causing a birth in the population.
I know that for $n$ individuals the time until the next birth is exponentially distributed with parameter $n(\lambda_1+\lambda_2)$ and likewise the time until the next death with parameter $n\mu$. The time until the next transition is the minimum of these two, therefore exponentially distributed with parameter $n(\lambda_1+\lambda_2+\mu)$. My question is if we can somehow distinguish whether a birth was caused by $\lambda_1$ or $\lambda_2$. So is there any way to get \begin{equation} \mathbb E_n (\text{ time until birth caused by } \lambda_1). \end{equation} It seems reasonable to say that the time is distributed exponentially with parameter $n\lambda_1$. Is this right?
This isn't really a mathematical question, just the result of a slightly sloppy use of language.
From the fact that you add $\lambda_1$ and $\lambda_2$ and call the result a "rate", we're led to conclude that $\lambda_1$ and $\lambda_2$ are real numbers. But then this talk about $\lambda_1$ and $\lambda_2$ causing births (or being ways of causing a birth) doesn't make any sense when taken literally: No matter how Platonist you are, real numbers don't cause births.
Most people reading this will probably resolve this tension by (explicitly or implicitly) thinking that what you actually mean is that there are two types of birth (or of causing birth) and that $\lambda_1$ and $\lambda_2$ are the respective rates at which they independently occur with exponential distribution. But then no question remains, because the answer to the question you pose at the end of the post is then "yes" by definition.
So either the question is answered trivially, or, if you mean something else by this talk of $\lambda_1$ causing births, you'd have to define that before the question could be answered.