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Let us now take
stock and summarize

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what we have done for these
two processes, the Bernoulli

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and the Poisson process,
and their relation.

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The Poisson process
runs in continuous time,

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whereas, for the
Bernoulli process,

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time is discrete and
is divided into slots.

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The Poisson process is defined
by a single parameter, lambda,

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which is the intensity
or arrival rate,

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and tells us the expected number
of arrivals per unit time.

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For the Bernoulli
process, we have, again,

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one basic parameter, which
is the probability of success

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at any given trial,
or the probability

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of an arrival during
each one of the slots.

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Based on our model, we were
interested in three kinds

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of quantities.

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And we found the
distributions of them.

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The first quantity is
the number of arrivals

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during a certain time interval.

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For the discrete case,
the number of arrivals

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has a binomial distribution,
whereas for the one Poisson

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case, the distribution is that
of a Poisson random variable.

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Then we looked at the time
until the first arrival,

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or the time between
consecutive arrivals.

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For the Bernoulli process,
that distribution is geometric.

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For the Poisson process, that
distribution is exponential.

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Note that in this
instance, we're

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dealing with a discrete
random variable,

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but, here, with a
continuous random variable.

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And then, as a
generalization, we

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could find the time until a kth
arrival, which, in the Poisson

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case, is given by an
Erlang distribution.

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And for the Bernoulli
case, we developed

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one particular formula.

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And that formula
is actually known

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under the name of the
Pascal distribution.

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All of these results,
for the Poisson case,

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were obtained because we used
an approximation argument.

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That is, we had the results
for the Bernoulli case.

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But then we argued that
the Poisson process

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is a limiting case of
the Bernoulli process

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in which we take time, divide
it into a large number of slots,

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during each one of
the slots, however,

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we have a small
probability of an arrival.

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And this is done in a way so
that the product of these two

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numbers stays a constant.

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By using a finer and
finer discretization,

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we could approach the
Poisson process arbitrarily

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close by a Bernoulli process.

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And then we used the
Bernoulli formulas

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in which we took the limit
as delta was going to zero.

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And this gave us the result
for the Poisson case.