When p is small and N is large, then the Bi(N,p) random variable can be well approximated by a Poisson Random Variable with Mean lambda equal to Np.
The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore, it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. There is a rule of thumb stating that the Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10.
The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". If the probability p is small, then the probability of two independent occurrence is the square of p, which will be MUCH smaller, and can often be ignored. For example if the probabilty of one occurrence is p=0.01 or 1%, then the probability of two occurrences is 0.0001 or 1 in 1000, which is negligible.
With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. Let this total number be
. Divide the whole interval into
subintervals of equal size, such that > (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in an interval
for each is equal to . Now we assume that the occurrence of an event in the whole interval can be seen as a Bernoulli trial, where the trial corresponds to looking whether an event happens at the subinterval
with probability . The expected number of total events in such trials would be , the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form
. As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as
goes to infinity. In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem.