L22.4 The Poisson PMF for the Number of Arrivals
The definition of the Poisson process gives us information about the probability that we get k arrivals during an interval of length delta when delta is a very small number. How can we find the probability of k arrivals during an interval of some general length tau, where tau is no longer a small number? In particular, we're interested in the random variable denoted N sub tau, which stands for the number of arrivals during an interval of length tau. And we wish to find the PMF of this random variable, the probability that N sub tau is equal to k, and which is what we have been denoting by this particular notation in the context of the Poisson process. Now if, instead of the Poisson model, we had for the Bernoulli process model, we would know the answer. S, the number of successes, or number of arrivals in n slots, has a PMF which is given by the binomial formula. Can we somehow use what we know about the Bernoulli process to find the answer for the Poisson process? Th...