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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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00:00:10,500 --> 00:00:12,700
we can go ahead and
try to calculate

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the mean and variance
of this quantity.

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00:00:15,540 --> 00:00:18,630
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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00:00:46,790 --> 00:00:50,850
is well approximated by a
binomial random variable

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00:00:50,850 --> 00:00:55,210
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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00:01:12,550 --> 00:01:15,130
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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00:01:50,200 --> 00:01:54,009
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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00:02:04,390 --> 00:02:06,610
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.