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In this lecture, we discuss
the simplest nontrivial

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stochastic process, the
Bernoulli process.

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Time is divided into slots.

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At each slot, we have
an independent trial

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such as a coin flip.

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And each trial results
in heads or tails.

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Or in different language, each
trial may or may not result in

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an arrival or a success.

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Pretty much everything we
will do will be a simple

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application of skills that
we already have.

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For example, we already know
that the number of arrivals in

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n time slots is described
by the binomial PMF.

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We will then discuss some
consequences of the

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independence of the
different trials.

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Basically, the process
has no memory.

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Whatever happens in the future
is not affected by whatever

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happened in the past.

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By exploiting this property, we
will find the distribution

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of the time of the first arrival
and more generally of

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the time of the kth arrival.

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We will also find the
distribution of the time

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between consecutive arrivals.

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Next, we will take two
independent Bernoulli

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processes, let's say arrivals of
men and arrivals of women,

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and merge them to get an overall
arrival process.

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We will see that the
merged process is

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also a Bernoulli process.

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We will also look at the reverse
operation, namely,

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splitting a Bernoulli process
into two separate processes.

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Finally, we will look at a
particular asymptotic regime

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in which we have a large number
of time slots but a

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very small probability of an
arrival during each time slot.

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We will carry out some algebraic
manipulations.

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And we will see that the
binomial PMF for the number of

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arrivals can be well
approximated by the so-called

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Poisson PMF.