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!
Some hints:
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 $t$.
One point to note that what is constant is the expected number of deaths per unit of time. Some other people use "constant death rate" to mean that the expected number of deaths per unit of time per unit of population, i.e. what you describe as linear.
