Let X1, X2, ... be a sequence of independent and identically distributed random

Question

Let X1, X2, ... be a sequence of independent and identically distributed random variables with distribution [ i.e. CDF ] F, having a finite mean and variance. Whereas the central limit theorem states that the distribution of

SUM_{i=1 to n} Xi

approaches a normal distribution as n goes to infinity, it gives us no information about how large n need be before the normal becomes a good approximation. Whereas in most applications, the approximation yields good results whenever n is at least 20, and oftentimes for much smaller values of n, how large a value of n is needed depends on the distribution of Xi. Give an example of a distribution F such that the distribution of

SUM_{i=1 to 100} Xi

is not close to a normal distribution.

Explanation / Answer

There are many distributions that do not follow normal distribution. The sum of such data points do not follow normal distribution , even if the sample size is large enough. For example : Beta distribution it applies to random variables limited to a fixed interval. So it cannot follow normal distribution no matter what the sample size be.

Likewise, logistic distribution & gamma distributions are a few set of distributions that are non normal.

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