WEBVTT
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Now that we have in
our hands the PMF
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of the random variable N tau,
which is the number of arrivals
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during an interval
of length tau,
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we can go ahead and
try to calculate
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the mean and variance
of this quantity.
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Regarding the mean, we could
use just the definition
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of the expected value
and then carry out
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of this calculation,
which is not too hard.
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And in the end, we would obtain
an answer equal to lambda times
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tau.
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This is indeed the correct
formula for the expected value.
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But let us understand
why this formula should
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be true without doing
any calculation.
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Remember that the random
variable, the number
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of arrivals in the
Poisson process,
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is well approximated by a
binomial random variable
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with those particular
parameters n and p in the limit
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when delta goes to zero.
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And this works through a
discretization argument.
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Therefore, the
expected value of N tau
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should be approximately equal
to the expected value of that we
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get from the Bernoulli
processes, that is the expected
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value of the binomial
random variable.
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And the expected value of
a binomial random variable
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is n times p.
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And n times p evaluates
approximately to lambda times
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tau.
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The second equality here is
approximate because we're
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neglecting this order
of delta squared term.
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Now, these approximations
are increasingly
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exact as we let delta go to 0.
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And when we take the
limit as delta goes to 0,
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we see that the expected value
of the number of arrivals
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in the Poisson process will
be equal to lambda tau.
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For the variance, we can
follow a similar argument.
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First do a binomial
approximation
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and use the formula
for the variance
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of a binomial random variable.
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And then, when delta is
small, this number p is small.
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And it's negligible
compared to 1.
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n times p is approximately
lambda [tau].
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And so we obtain this expression
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This expression here
is the correct one.
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If we were to use the formal
definition of the variance
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and carry out the
calculations using the PMF,
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this is what we would
obtain, except that it
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would be somewhat tedious.
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The formulas that
we have derived,
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at least the first
one, is quite intuitive
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and has a natural form.
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The expected number of arrivals
is proportional to tau.
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If we double the length of the
time interval for interest,
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we expect to see twice
as many arrivals.
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This formula also reinforces
the interpretation of lambda
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as an arrival rate.
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Since lambda is the expected
number of arrivals divided
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by the length of
time, it is, really,
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the expected number of
arrivals per unit time.
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So it's natural to call
lambda the arrival rate,
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or the intensity of
the arrival process.
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Finally, regarding the
variance, it is a curious fact
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that the variance
turns out to be
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exactly the same as
the expected value.
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And this is a special
property of the Poisson PMF.