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Let us now look at some
numerical examples to get a

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feel for how the central limit
theorem works in practice.

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Let us look at a discrete random
variable that has a

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uniform distribution in the
range here from 1 to 8.

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If we add two independent random
variables, drawn from

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this PMF, we obtain a random
variable whose PMF is the

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convolution of this
PMF with itself.

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We can even carry out this
calculation by hand, and we

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get a triangular PMF.

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So this is what we get for the
case where n is equal to 2.

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Now we can keep doing this.

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If we add four of these discrete
uniforms, of course

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assumed independent, then we
obtain a PMF that starts to

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have a shape close to that
of a normal shape.

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And if you take as many as 32
of them, then the PMF of the

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sum of 32 discrete uniforms is
almost identical to the shape

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that you would get if you were
to draw a normal PDF.

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So with n as small as 32, we
have essentially converged.

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In fact, this convergence is
so good that in practice,

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quite often people use this
idea to generate random

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samples of a normal
random variable.

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What do you do?

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You draw 32 random samples
from a discrete

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uniform, add them up.

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And what you get is a sample
of essentially a

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normal random variable.

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Now in this example, things
worked out well for us because

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the distribution that we
started from was nicely

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symmetric and didn't have
any strange features.

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Things are not always
so favorable.

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Let us consider starting from
a truncated geometric.

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If we add eight random variables
that are independent

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and drawn from this
distribution, what we obtain

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is a PMF of this form, which
does not really look like a

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normal shape.

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The reason is that there's
a pronounced asymmetry.

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So let us add more and
more independent X's.

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If we add 16 of them, we start
to get something that's a

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little closer to normal.

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But the asymmetry is
still visible.

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And if we add 32 of them, we can
still see some asymmetry.

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Namely, this tail here does
not look exactly like this

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tail out there.

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So in this instance, it's really
the asymmetry of the

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original distribution
that makes it

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difficult to converge.

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And it will take larger values
of n before we can get a very

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accurate approximation.