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We now start our discussion
of stochastic processes

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by starting with the simplest
stochastic process there is.

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This is the so-called
Bernoulli process,

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which is nothing but a sequence
of independent Bernoulli

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trials.

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We let Xi stand for the
random variable that

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describes the result
in the i trial.

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The assumptions that we
make are that at each trial

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there is a certain probability,
p, that the trial results in 1.

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And in that case, we
usually say that there

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is a success at the ith trial.

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And the remaining
probability, 1 minus p,

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is assigned to the possibility
that the random variable Xi

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takes a value of
0, in which case

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sometimes we say that
there was a failure.

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Now, to keep things
nontrivial, we

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will always assume that p is
a number strictly between 0

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and 1, because otherwise, in the
extreme cases of p equal to 0

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or p equal to 1, there
isn't really any randomness.

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This process is something
that we have already seen.

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We have worked plenty
of examples involving

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repeated Bernoulli trials
or repeated coin flips.

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We have solved several problems,
we have seen several formulas.

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Here we will recapitulate
some of them.

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But we will also start
looking at the process

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from a new point of view.

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Before continuing, let me
emphasize the assumptions that

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come into the Bernoulli process.

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One key assumption is
that we have independence.

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The different trials
are independent.

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The second assumption
that we make

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is that the model
is time homogeneous.

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What I mean by this is
that this probability

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p of success at a given trial
is the same for all trials.

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It does not depend on i.

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So this process is as simple as
a stochastic process could be.

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But nevertheless, it
can be used as a model

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in various situations.

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Sometimes it's clear that
we're dealing with a Bernoulli

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process, but sometimes it also
shows up in unexpected ways.

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In any case, the first simple
example could be the following.

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Every week you play the
lottery, and either you win

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or you do not to win.

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And assuming that it is
the same kind of lottery

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that you play each
week, the constant p

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would be the same, the
probability of success.

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And assuming a lottery that
is not rigged in any way,

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whether you win on
one week, should

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be independent from what
happens in other weeks.

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Quite often, the
Bernoulli process

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is used as a model of
arrivals, in which case,

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instead of saying
probability of success,

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we would say probably
of an arrival.

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The idea is that
time is slotted,

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let's say, in
seconds, for example.

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And each second you're sitting
at the entrance of a bank

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and you make a note
whether somebody

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came into the
bank, in which case

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we have an arrival or success,
or whether no one came

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during that time interval.

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If one here believes
that different slots,

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different seconds, are
independent of each other,

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then you do have a
Bernoulli process.

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It might not be an
exactly accurate model,

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but it is a good
first approximation

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to start working with a
model of this situation.

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Similarly, if you have a
server, a computer, that

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takes jobs to process and
jobs are coming randomly,

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you divide time into
slots, and during each slot

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a job might arrive
or might not arrive.

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And as a first approach
to a model of this kind,

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you might as well employ
the Bernoulli process.

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A final note, why is this
process called the Bernoulli

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process?

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Well, the name comes
from a famous family

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of Swiss mathematicians,
the Bernoulli family.

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And one of them,
Jacob Bernoulli,

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did many contributions to
many branches of mathematics.

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But an important
one was in the field

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of probability,
where he actually

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derived quite deep results
on a sequence of what we now

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call Bernoulli trials.