Holding Time for Birth and Death Process

Question

My text claims that, if we have a birth and death chain, with arrivals at an exponential rate $\lambda$ and departures at an exponential rate $\mu$, then the time until either a birth or death is exponential with rate $\lambda + \mu$. Why?

Answer 1

Let $X$ be exponentially distributed with parameter $\lambda$, and let $Y$ be exponentially distributed with parameter $\mu$. Assume $X$ and $Y$ are independent. This is somewhat unreasonable.

The time until a birth or death is the minimum of $X$ and $Y$. Call this random variable $Z$. We find the cumulative distribution function $F_Z(z)$ of $Z$, and then the density function of $Z$.

We have $Z\gt z$ if and only if $X\gt z$ and $Y\gt z$. This has probability $e^{-\lambda z}e^{-\mu z}$. Thus $$F_Z(z)=1-e^{-(\lambda+\mu)z}$$ (for $z\gt 0$). For the density function of $Z$, differentiate. We get $$f_Z(z)=(\lambda+\mu)e^{-(\lambda+\mu)z}$$ (for $z\gt 0$), the familiar exponential density function.