Central Limit Theorem Illustrated with Four Probability Distributions
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We consider four different probability distributions for 
as a function of 
between the interval 
and 
; outside this interval, the probability is set to 0. The are 
, 
, 
and one that is discrete; 
is the constant that enables the integral of the probability distribution to be normalized; that is, 
.
Contributed by: D. Meliga and S. Z. Lavagnino (May 2018)
Additional contribution by: G. Valorio
Open content licensed under CC BY-NC-SA
Snapshots
Details
Snapshot 1: a small 
and small 
result in a poor approximation of a Gaussian; this is also shown by the discrepancy between estimated and real values
Snapshot 2: a large and small result in a poor approximation of a Gaussian, but an accordance between estimated and real values can be found
Snapshot 3: a small and large result in a good approximation of a Gaussian, and the estimated and real values are close to each other
Snapshot 4: a large and a large result in a very good approximation of a Gaussian, and the estimated and real values are close to each other
Reference
[1] R. E. Walpole, R. H. Myers, S. L. Myers and K. Ye, Probability & Statistics for Engineers & Scientists, 9th ed., Boston: Prentice Hall, 2012.
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