I have a pure death process $X=\{X(t) : 0 \leq t < \infty\}$ with parameters $\lambda_n=0$ and $\mu_n=\mu$ and if $X(0)=N$ and I'm supposed to determine $P_n(t)=P\{X(t)=n\}$ for $n=0,1,2,\ldots,N$
So I'm supposed to find $\Pr(X(t)=n\mid X(0)=N)$. I beleive this is a Poisson process to find the probability that exactly $N-n$ deaths have happened by time $t$, is that correct? How would I express this probability?
Best Answer
Let $Y_t$ be a Poisson process with rate $\mu$. Since $\mu_n = \mu$, we can take the deaths as following $Y_t$ until everyone has died. So given you start with $N$ people, and $n \ge 0$, the probability that $X(t) = n$ is the probability that there are exactly $N-n$ occurrences in the $Y_t$ process in the interval $[0,t]$. The number of such occurrences in an interval of length $t$ is a Poisson random variable with parameter ..., so its probability of taking the value $N-n$ is ....