I am trying to grapple with the following question as I self-study from the chapter on probability in my quantum mechanics textbook (Ballentine's Quantum Mechanics: A Modern Development). Unfortunately, it has been a long time since I've studied probability and my memory is hazy.
A source emits particles at an average rate of $\alpha$ particles per second; however, each emission is stochastically independent of all previous emission events. Calculate the probability that exactly $n$ particles will be emitted within a time interval $t$.
Now I have the vague notion that I should be able to situate this question in the context of a Poisson process but I'm not quite sure how to go about it. In particular, all I can deduce so far is the following:
We interpret the problem statement as saying that the probability of an emission in any infinitesimal interval $dt$ of time is $\alpha \ dt$. Then we can argue that the probability $P_1$ of the first emission happening in the interval $[t,t+dt)$ is the probability of the event in which no emissions happen in the interval $[0,t)$ and we have an emission in the interval $[t,t+dt)$. That is, we have $$P_1 = \left(\lim_{N \to \infty} \left(1-\frac{\alpha t}{N}\right)^N\right)\left(\alpha \ dt \right) = \alpha e^{-\alpha t}dt := f(t)dt$$ where $f(t)$ is the corresponding pdf (and we've just shown that it's exponential).
It's not clear to me how to generalize this statement about the random variable $T_1$ (time until first emission) to the pmf/pdf of the random variable $n_t$ (the number of emissions in a time interval $t$). To use the theory of point Poisson processes I think I need to show the definition: "a Poisson point process has the property that the number of points in a bounded region of the process's underlying space is a Poisson-distributed random variable." But it's not clear to me how I can use my $f(t)$ (for the RV $T_1$) to show this fact. Any help with showing this fact (and ideally basically helping me work through the whole problem) would be greatly appreciated.
