# Death process (stochastics)

From what I understand, the question is asking me to find P(X(t) = n | X(0) = N). I know that with a linear death rate this probability is (N choose n) * [e^(-alpha*t)]^n * [1 - e^(-alpha*t)]^N-n but I don't think this is true for a constant death rate. Any help on how to approach this would be great! Also, I see in the final solution that there are two answers: one for n = 1, 2, ... N and another one for n = 0. Why is this case? And how would I go about finding both cases? Thanks!

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This seems to be Example 3 here – Henry Mar 19 '11 at 11:53

To find $P_n(t) = Pr(X(t)=n|X(0)=N)$ you need to find the probability that exactly $N-n$ deaths have happened by time $t$. This looks Poisson with a suitable parameter.
The reason $n=0$ has a different form is that there is an unstated assumption that the population cannot fall below $0$, despite the literal implications of an indefinitely constant death rate, so you are looking for the probability that at least $N$ deaths have happened by time $t$.