WEBVTT
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We have seen that
under some conditions,
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the binomial PMF is well
approximated by a Poisson PMF.
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But we have also seen
the central limit theorem
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that tells us that the binomial
PMF can be approximated
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using a normal random variable.
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Can we reconcile
these two facts?
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Let's look into the situation
in some more detail.
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Consider a Poisson process
that has rate equal to 1.
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And consider that
Poisson process
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running over the
unit time interval.
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We take the unit
interval, and we split it
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into many small
sub-intervals, where
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each sub-interval
has a length of 1/n.
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And let Xi be the number of
arrivals that we get during
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the i'th interval.
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Xi is a Poisson random variable.
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And the mean of that
random variable,
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or the parameter of
that random variable,
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is just the duration of the time
interval, since the rate is 1.
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So it's a Poisson random
variable, with parameter 1/n.
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Now, let us look at the
total number of arrivals.
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The total number of arrivals
is the sum of how many arrivals
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we had during each one
of these intervals.
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And we know the
distribution of S. S
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is a Poisson random variable,
with parameter equal to 1.
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Now, here what we have is
a sum of random variables
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that are independent and
identically distributed.
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They are identically
distributed,
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because all of these intervals
have the same length.
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And they're independent,
because in the Poisson process,
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what happens in
different intervals are
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independent events.
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So we are in a
situation where we
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could apply the
central limit theorem.
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We have a sum of
many independent,
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identically distributed
random variables.
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And by letting n go to infinity,
the central limit theorem
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appears to tell us that
S is going to be normal.
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Now, how can we reconcile
these two facts?
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We know that the Poisson
distribution is not the same
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as a normal distribution.
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What is the catch?
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Well, the catch
is the following--
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the central limit theorem
applies to a situation where we
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fix a certain
probability distribution,
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the distribution of the Xi's.
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And it tells us that as we add
more and more of these Xi's,
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asymptotically, we obtain
a distribution that's
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well approximated by a normal.
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On the other hand, what we have
here is actually different.
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The Xi's do not have
a fixed distribution.
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But rather, the distribution
of Xi depends on n.
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That is, if we change
n so that we're
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adding more random
variables, we're
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adding more random
variables that are now
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coming from a
different distribution.
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And this is not a situation
to which the central limit
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theorem applies.
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And therefore, this conclusion
here is not justified.
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And so there's no
contradiction between the two
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types of approximations.
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To summarize, the
situation is as follows.
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Consider a binomial random
variable with some parameters
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n and p.
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Now, let p be fixed.
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And let n go to infinity.
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In that case, the
binomial random variable
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can be thought of as the sum of
n Bernoulli random variables.
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And those Bernoulli
random variables
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have a parameter
p, which is fixed.
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So we're dealing with the
sum of iid random variables
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from a fixed distribution.
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And this is the situation
where the central limit
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theorem applies.
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And we have a normal
approximation.
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On the other hand, if
we take the product
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n times p, which is the
expected value of this binomial,
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to stay constant,
but we let n go
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to infinity and at the same
time let p go to 0, then
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in this regime, in the
limit, this random variable
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will be well approximated by
a Poisson random variable.
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So we have two different
approximations.
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Both of them are
valid, but they're
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valid in different regimes.
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Now, although they're
different, there's
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actually an interesting case in
which the two will not really
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differ.
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And this is the following.
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Consider a Poisson random
variable with parameter n.
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And we're interested in the
limit as n goes to infinity.
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We can think of
this random variable
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as the number of arrivals
during an interval
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of length n in a Poisson process
with arrival rate equal to 1.
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Now, let's take this
interval and split it
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into n intervals, each of
which has a length of 1.
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And let us call Xi the number of
arrivals in the i'th interval.
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Our Poisson random variable
is going to be, of course,
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equal to the sum of
the number of arrivals
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during each one
of the intervals.
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Each one of these
random variables
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is Poisson with
parameter equal to 1.
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And these random variables
are actually iid.
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Now, what's happening
in this case
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is that even when we
increase n, because we're
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using constant length intervals,
the distribution of the Xi's
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doesn't change.
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So this is a situation
in which we're
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going to get approximately a
normal random variable as n
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goes to infinity.
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So what we see is that a
Poisson random variable, but
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with a very large
parameter, starts
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to approach the
normal distribution.
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And this in particular
will tell us
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that these two approximations
that we have, in some regime,
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they would start to agree.
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Now, all of this discussion
here has been asymptotic.
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We talk about p going to
0 or n going to infinity.
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But in any real situation, you
will be given actual numbers.
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And you cannot really tell,
is this number close to 0,
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or is it not?
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Here, we need some rules of
thumb or maybe some experience.
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Let's look at some examples.
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In this case, n times
p is equal to 1.
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So the number of
arrivals or the values
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of the binomial random variable
will take values 0, 1, 2,
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3, but with high probability,
not a lot more than that.
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So the binomial random
variable is really
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a discrete random variable.
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There's no way to
approximate it with a normal.
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On the other hand,
p is very small.
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So a Poisson
approximation would be
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very reasonable
in this situation.
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On the other hand, if
p is equal to 1/3, then
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definitely 1/3 is
not a small number.
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A Poisson approximation
would not apply.
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But n is pretty big.
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So that a normal approximation
would be appropriate.
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And finally, in this case,
we would obtain a Poisson
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approximation with parameter
100, because n times p is 100.
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But a Poisson random
variable with parameter 100
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is also well
approximated by a normal.
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Or to think about
it differently,
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we start with a Bernoulli
distribution that's
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very skewed, [the] probability
of success is just 100.
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And this makes it difficult
for the central limit theorem
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to apply when you start with a
very asymmetric distribution.
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On the other hand, because
we're adding so many of them,
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the central limit theorem
actually does take hold.
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And so this is an example
where both approximations
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will be valid.
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So finally, to conclude, we have
two different approximations.
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They're valid in
different regimes.
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And in practice, you
need to do some thinking
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before deciding to choose one
versus the other approximation.