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Parameters of a
Addition of Independent Random Variables
Suppose
where the
are independent and they all have the same distribution with mean
and standard deviation
.
Then clearly
is also
.
Further, to find standard deviation, calculate
as all off diagonal terms vanish.
So it is precisely true that
y
has mean
and standard deviation
.
Later, we will show that as
(i.e., not exactly), all other moments of
y
vanish faster than
and so
y
becomes Gaussian whatever the original distributions of
x
.
Geoffrey Fox
,
Northeast Parallel Architectures Center
at Syracuse University,
gcf@npac.syr.edu
are independent and they all have the same
distribution with mean 
and standard deviation 
.
is also 
.
and standard deviation
.
(i.e., not exactly), all other moments of y vanish faster than 