Let us now take stock and summarize
what we have done for these two processes, the Bernoulli
and the Poisson process, and their relation.
The Poisson process runs in continuous time,
whereas, for the Bernoulli process,
time is discrete and is divided into slots.
The Poisson process is defined by a single parameter, lambda,
which is the intensity or arrival rate,
and tells us the expected number of arrivals per unit time.
For the Bernoulli process, we have, again,
one basic parameter, which is the probability of success
at any given trial, or the probability
of an arrival during each one of the slots.
Based on our model, we were interested in three kinds
of quantities.
And we found the distributions of them.
The first quantity is the number of arrivals
during a certain time interval.
For the discrete case, the number of arrivals
has a binomial distribution, whereas for the one Poisson
case, the distribution is that of a Poisson random variable.
Then we looked at the time until the first arrival,
or the time between consecutive arrivals.
For the Bernoulli process, that distribution is geometric.
For the Poisson process, that distribution is exponential.
Note that in this instance, we're
dealing with a discrete random variable,
but, here, with a continuous random variable.
And then, as a generalization, we
could find the time until a kth arrival, which, in the Poisson
case, is given by an Erlang distribution.
And for the Bernoulli case, we developed
one particular formula.
And that formula is actually known
under the name of the Pascal distribution.
All of these results, for the Poisson case,
were obtained because we used an approximation argument.
That is, we had the results for the Bernoulli case.
But then we argued that the Poisson process
is a limiting case of the Bernoulli process
in which we take time, divide it into a large number of slots,
during each one of the slots, however,
we have a small probability of an arrival.
And this is done in a way so that the product of these two
numbers stays a constant.
By using a finer and finer discretization,
we could approach the Poisson process arbitrarily
close by a Bernoulli process.
And then we used the Bernoulli formulas
in which we took the limit as delta was going to zero.
And this gave us the result for the Poisson case.