WEBVTT
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The definition of
the Poisson process
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gives us information
about the probability
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that we get k arrivals during
an interval of length delta
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when delta is a
very small number.
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How can we find the
probability of k arrivals
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during an interval of
some general length tau,
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where tau is no
longer a small number?
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In particular, we're interested
in the random variable denoted
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N sub tau, which stands
for the number of arrivals
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during an interval
of length tau.
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And we wish to find the PMF
of this random variable,
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the probability that N
sub tau is equal to k,
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and which is what we
have been denoting
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by this particular notation
in the context of the Poisson
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process.
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Now if, instead of
the Poisson model,
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we had for the
Bernoulli process model,
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we would know the answer.
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S, the number of successes, or
number of arrivals in n slots,
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has a PMF which is given
by the binomial formula.
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Can we somehow use what we know
about the Bernoulli process
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to find the answer for
the Poisson process?
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The answer is yes, and it
involves a limiting argument
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of the following kind.
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We take the interval
from 0 to t and divide it
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into a very large number of
intervals, so many of them,
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where each one of the intervals
has a length of delta,
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where delta is a small number.
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And to push the analogy
with the Bernoulli process,
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we will be calling those
little intervals as slots.
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Now during each slot, we may
get zero arrivals, one arrival,
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but there's also the
possibility that there
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may be two arrivals, or
even more than two arrivals,
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happening during
one of the slots.
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Because of this, the
picture that we have here
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is not quite the same as
for the Bernoulli process
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because in the Bernoulli
process, each one of the slots
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will get only 0 or 1.
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So the source of the discrepancy
between the two models
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is that here, a slot may
obtain two or more arrivals.
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But how likely is this?
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Let us look at the probability
that some slot, that
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is, any one of the slots,
contains two or more arrivals.
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That is, we're dealing with
the union of the events
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that slot i has two
or more arrivals.
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This event is the
union of these events
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and, therefore, the probability
of this event is less than
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or equal than the sum
of the probabilities
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of the constituents events.
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This is an inequality
that we have
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seen at some point in the past.
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And we're calling it
to the union bound.
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Now what is this summation?
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i ranges over the
different slots.
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And we have tau
over delta slots,
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so there's so many terms
that are being summed.
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Now, during any particular
slot, the probability
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of two or more arrivals is of
order delta squared, according
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to the definition of
the Poisson process.
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And this quantity
converges to 0 when
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we let delta become
smaller and smaller.
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So this means that the
discrepancy between the Poisson
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and the Bernoulli model, which
was due to the possibility
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that we might get two or
more arrivals during one
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of those slots, this
discrepancy is something
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that happens with
negligible probability.
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In other words, the
probability that we
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get k arrivals in
the Poisson model
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is approximately the
same as the probability
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that k slots have an arrival.
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Since we're neglecting
the possibility
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that some slot has
two or more arrivals,
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this means that the number of
arrivals in the Poisson model
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will be the same as the number
of slots that get an arrival.
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This approximate equality
becomes more and more exact
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as we let delta go to zero.
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But now what is this quantity?
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The probability that k
slots have an arrival
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is something that
we can calculate
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using the binomial
probabilities.
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Each one of the slots
has a certain probability
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of having an arrival.
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And different slots are
independent of each other
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by the defining properties
of the Poisson process.
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Therefore, this approximation
that we have developed
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satisfies the properties
of the Bernoulli process.
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We have a certain probability
that each slot gets an arrival.
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And we have independence
across slots.
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This means that we can
use now the PMF that's
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associated with
the Bernoulli model
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to calculate this quantity
and then take the limit,
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as delta goes to 0, to
obtain a formula for the PMF
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for the Poisson process.
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In more detail, what
we have is the number
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of arrivals, which
is approximately
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the same as the number of slots
that have an arrival, obeys
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a binomial distribution
in the limit as delta
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goes to 0-- a binomial
distribution in which
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the probability of arrival
during each one of the slots
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is approximately lambda
delta and the number of slots
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goes to infinity.
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And this happens in a way so
that the product of the two,
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n times p, is equal
to-- this term times
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this term gives us
a lambda times tau.
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This term times
this term gives us
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something that's order of delta.
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So it's negligible.
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So we have this
equality, and so this
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is approximately lambda
tau with the approximation
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becoming more and more exact
as we let delta go to zero.
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So all we need to do is to take
the formula for the Bernoulli
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process.
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Use these values of p
and n and take the limit.
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But this is a problem that
we have already encountered
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and have analyzed.
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If we let n go to
infinity, p goes to 0
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so that their product
stays constant,
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we have shown that the
binomial PMF converges
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to the so-called Poisson
PMF that takes this form.
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Notice one small
difference-- n times
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p here is equal to lambda,
whereas here, n times
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p is equal to lambda t.
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This means that we need
to apply this formula,
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but with lambda
replaced by lambda t,
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and this gives us
the final answer.
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This is the probability of k
arrivals during a time interval
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of lenght t in the
Poisson process.
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And this is a so-called Poisson
PMF with parameter lambda tau.
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To summarize, our strategy
was to argue that the Poisson
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process is increasingly
accurately described
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by a Bernoulli process
if we discretize
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time in a very fine
discretization.
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And the approximation
becomes exact in the limit
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when the discretization
is very fine.
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So we took the corresponding
binomial formula
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for the Bernoulli process
and took the limit
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to that's associated
with the parameters
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that we would obtain if we have
a very fine discretization.
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And this gave us
the final formula.