Let be a
series of independent random variables, each has an arbitrary
distribution. It is proved that the distribution of the normalized
variable
approximates a normal distribution as , where and are the mean and variance of
each random variable respectively.
Derivation
Once the distribution is decided, the mean and variance are both
constants. We let And aware
that monotonously
increase in the same order as (or
we say, ).
Consider the of the
normalized variable .
Apply the Inverse Fourier Transform to it, we obtain
The expression of
is written as Substitute in we obtain
Take every integral out with the dummy variable written as . Then substitute in the result for every integral
in (5), we obtain Then we just need to calculate from the inverse function.
We have This is exactly the form of the to a normal distribution. Hence,
we can conclude that
follows a normal distribution.