Central Limit Theorem for the Continuous Uniform Distribution

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This Demonstration illustrates the central limit theorem for the continuous uniform distribution on an interval. If has the uniform distribution on the interval
and
is the mean of an independent random sample of size
from this distribution, then the central limit theorem says that the corresponding standardized distribution
approaches the standard normal distribution as
. Using operations on the characteristic function of
we can compute the PDF of
more easily than we could directly. The blue curve is the PDF of
and the dashed curve is the PDF of a standard normal distribution.
Contributed by: David K. Watson (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
If is the PDF of a random variable
, then
,
, (the convolution of
and
). This means that finding the PDF of
involves computing the
-fold convolution of
with itself, a computationally intensive operation to do directly even for small
and simple PDF functions. This can be made easier in some cases by using the convolution property of Fourier transforms,
, and the observation that the characteristic function of
is the same as the Fourier transform
, which means that
can be computed by rescaling the inverse Fourier Transform of
. For the uniform distribution, this lets us compute the distribution for
much more quickly than we could directly, though it is still a little slow for larger
.
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