Problem:
The birth and death process with parameters $\lambda_n=0$ and $\mu_n=\mu, n>0$ is called a pure death process. Find $P_{i j}(t)$.
Solution:
Since the death rate is constant, it follows that the number of deaths evolves as a Poisson process until the system becomes empty. Hence, for $i \geq j>0, P_{i, j}(t)$ is the probability that a random variable having a Poisson distribution with parameter $\mu$ t the value $i-j$, i.e., $$ P_{i, j}(t)=e^{-\mu t} \frac{(\mu t)^{i-j}}{(i-j) !}, $$ and for $i \geq j=0$, it is the probability that values of $i$ or larger are assumed. $$ P_{i, 0}(t)=1-\sum_{k=0}^{i-1} e^{-\mu t} \frac{(\mu t)^k}{k !} $$
Confusion:
For the first equation where $j>0$ I understand where the first equation comes from. My confusion comes from the last equation. I try to understand this formula by focussing on the complement part, aka the summation part. The summation part denotes the events that you land in state ${i,i-1,.....,1}$. So rewriting the formula results in:
\begin{equation} P_{i, 0}(t)=\sum_{k=i}^{\infty} e^{-\mu t} \frac{(\mu t)^k}{k !} \end{equation}
But in the case that $k > i$, the number of "deaths" exceeds the number you started at, so in all these cases, the system was already exhausted, which yields a probability 0.
This gives you:
\begin{equation} P_{i, j}(t)=e^{-\mu t} \frac{(\mu t)^{i}}{(i) !} \end{equation}
This equation is effectively the same as the first at the top of the post, but only with $j=0$
Question
- Did I made an error in my reasoning? If yes, could I please get feedback on that?
- If not, why did the authors deemed it necessary to intoduce a new equation? It seems a bit trivial to introduce a new equation while the first one is sufficient (which makes me suspicious that I made an error).
Tim
This is still a birth-death process; the death rate being a constant $\mu$ means that only transitions from states $i$ to $i-1$ occur for $i\geqslant 1$, and the times are exponentially distributed with rate $\mu$. So we have $$ P_{i,j}(t) = \begin{cases} 0,& j>0,j\ne i-1\text{ or } (i,j)=0\\ \mu e^{-\mu t},& j=i-1\text{ and } j\geqslant 1. \end{cases} $$