WEBVTT
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We have seen that the
binomial distribution plays
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an important role in the study
of the Bernoulli process.
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And the reason is that the
binomial distribution describes
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the number of arrivals during
a fixed number of slots.
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We will now develop
an approximation
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to the binomial
distribution that
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applies to one
particular regime,
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and that regime is when we have
a very large number of slots,
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but we have a small probability
of success in each slot.
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And this is in a way so that the
product of these two numbers,
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which is the expected
number of successes,
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is a moderate number.
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One example where such
a situation might arise
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is the following.
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Suppose you're interested
in earthquakes in your city,
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and you divide time
into slots of one hour.
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During each hour,
the probability
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of having a noticeable
earthquake in your city
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would be a very small number.
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On the other hand, if
you're interested in a time
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frame of five
years, there's going
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to be many hours
during that time frame,
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so that n would be quite large.
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But the expected
number of earthquakes
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over a period of five years
should be a moderate number.
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And one can think
of other situations
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where this regime might arise.
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The one particular
situation that
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will be very
interesting for us is
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going to be when we try
to take a continuous time
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approximation of the
Bernoulli process
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by dividing time into
very small slots,
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so that we have many slots, but
a small probability of success
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during each one of those slots.
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So to start, let us look at
the form of the binomial PMF.
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And let us just try to develop
an approximation to this PMF,
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when we fix k to be
particular constant number,
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and then take the limit as
n goes to infinity and p
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goes to 0, but in a way that
lambda remains constant.
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And in particular,
because of this relation
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here, we will have p
equal to lambda over n.
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So let us take this expression
and start rewriting it.
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Let us look at the ratio of
n factorial divided by this.
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The denominator has the
product of all numbers going up
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to n minus k.
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So by dividing by this number,
what is left out of the n
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factorial is only the terms
that go up to n minus k plus 1.
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Then we have, in the
denominator, the factor
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of k factorial.
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Now p is equal to lambda over
n, so this term becomes lambda
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to the k divided by n to the k.
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And similarly,
for the last term,
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we have 1 minus lambda over
n to the power n minus k.
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Now let us rearrange terms.
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Here, we have a product of
k terms in the numerator.
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Here, we have n
multiplying itself k times.
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So we can take a factor
of n and place it
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underneath each
one of those terms
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to obtain n over n times
n minus 1 over n times--
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we continue this way all the
way to n minus k plus 1 divided
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by n.
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Take this term, k
factorial, move it
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underneath the
lambda to the k term,
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and then let us
split this last term
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into 2 pieces in this manner.
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And now let us start taking
limits as n goes to infinity.
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The first term that we
have here is equal to 1.
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How about the second term?
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n divided by n is equal
to 1, 1 over n goes to 0,
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so this term also
converges to 1.
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And by a similar argument, all
of the terms in this product,
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including this
one, converge to 1.
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The term lambda k
over k factorial
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remains exactly as is.
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And now, let us
look at this term.
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This is probably familiar.
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There is a basic
fact which tells us
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that if we take this
expression and raise it
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to the nth power, what
we get is e to the minus
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lambda in the limit
as n goes to infinity.
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So using this basic
result, this term
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becomes e to the minus lambda.
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And finally, let's
look at the last term.
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Remember that k is
fixed, is a constant.
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1 minus lambda over
n converges to 1,
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and when we raise that
number to the k-th power,
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we still get a 1 in the limit.
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So the only terms that
are left are here,
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and essentially,
what we have just
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established is
that in the limit,
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the probability of k arrivals
in a Bernoulli process
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or the binomial
probability evaluated
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at k, in the limit, as n goes
to infinity and p goes to 0,
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is given by this formula, here.
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This is the formula
for the Poisson PMF.
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And so what we have established
is that the binomial PMF
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converges to a
Poisson PMF when we
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take the limit in
this particular way.