Illustrating the Central Limit Theorem with Sums of Bernoulli Random Variables

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Consider the central limit theorem for independent Bernoulli random variables , where
and
,
. Then the sum
is binomial with parameters
and
and
converges in distribution to the standard normal. The exact distribution for
may be written
, where
,
,
. The convergence may be illustrated using rectangles of width
and height
centered at
. As
increases, the rectangles closely approach the standard normal density function. The convergence is faster in the symmetric case when
.
Contributed by: Ian McLeod (University of Western Ontario) (February 2010)
Open content licensed under CC BY-NC-SA