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We now set out to study
the Poisson process, which

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is a continuous time version
of the Bernoulli process.

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In the Bernoulli process,
time is divided into slots.

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And during each
one of the slots,

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we may either have an
arrival or no arrival.

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In the Poisson process,
time is continuous.

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And we may get
arrivals at any time.

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We want to define
the Poisson process

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by introducing some assumptions
that in some ways parallel

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the assumptions that we made
for the Bernoulli process.

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What where those assumptions?

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The first one we made was the
assumption of independence--

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namely that what happens
in different slots

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are independent.

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Similarly, for the
Poisson process,

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we will make the following
independence assumption.

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If we consider two intervals,
two time intervals that

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are disjoint, and look at
the random variable that

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stands for the
number of arrivals

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during this interval
and that interval,

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we will assume that these
two random variables

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are independent.

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But even more than
that, if we take

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any collection of
disjoint time intervals,

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and we look at the
associated random variables,

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the associated
numbers of arrivals,

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that collection
will also consist

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of independent random variables.

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The second assumption for
the Bernoulli processes

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was one of time homogeneity,
namely at each slot,

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we had the same
probability of an arrival.

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We want to make a similar
assumption for the Poisson

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process, and it's going
to be the following.

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The probability that
we have k arrivals

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during an interval of
a certain duration tau

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is going to be the
same no matter where

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that interval sits.

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So if this is an interval
that has a certain duration,

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and this is an interval
that has the same duration,

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the probability of three
arrivals in this interval

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is going to be the same as
the probability of three

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arrivals in that interval.

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And therefore, since
this probability does not

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depend on where
the interval sits,

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that probability
will be fully defined

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by the number of arrivals
that we're interested in

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and the length of
the interval as

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opposed to the location
of the interval.

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So we will be using
this notation here

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to indicate this probability.

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In this notation, we think
of tau as a constant.

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And then, P of k corresponds
to probabilities.

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In particular, if
you add over all k's

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the various probabilities,
what you should get

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would be a value
of 1, because this

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exhausts all the possibilities.

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And k here ranges
from 0 to infinity,

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because we allow an arbitrarily
large number of arrivals

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during a given time interval.

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Now, with this
assumption in place,

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it would still be
possible to have

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arrivals that happen
simultaneously,

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multiple arrivals at
the same time point.

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In order to avoid
this situation,

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we introduce one more assumption
which is the following.

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It talks about the
number of arrivals

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during a time interval that
has a small length delta.

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During a small time
interval, there

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is negligible probability of
having more than one arrival.

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We will either have
one or zero arrivals.

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And the probability
of one arrival

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is a certain number,
lambda times delta.

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It's proportional to delta.

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So if the interval becomes
smaller and smaller,

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that probability also goes to 0.

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But it goes to 0 at a rate
proportional to delta.

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So you can think of lambda
as probability per unit time.

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These are the units of lambda.

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Now here, I'm writing
an approximate equality.

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What does that mean?

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It means that these are
not exact expressions.

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But they are exact within
a second order term.

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And a second order
term is a term

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that's negligible compared
to the first order term

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when delta is small.

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More precisely,
mathematically, what we mean

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is that a second order term
compared to a first order term

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goes to 0 as delta goes to 0.

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Finally, let me
reiterate that lambda

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should be interpreted
as an arrival rate.

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It is a probability
per unit time.

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The bigger lambda is, the
bigger the probability

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is that we get an arrival
during a small time interval.

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If we double lambda, then
we double the probability

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that we have an arrival
during a small time interval.

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And so we expect to have about
twice as many arrivals, hence

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the interpretation
as an arrival rate.

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We will see shortly that
this is also justified

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because lambda shows up in
expressions for the expected

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number of arrivals
during a time interval.