Amazingly, the distribution of a sum of two normally distributed independent variates 
and 
with means and variances 
and 
, respectively is another normal distribution
![P_(X+Y)(u)=1/(sqrt(2pi(sigma_x^2+sigma_y^2)))e^(-[u-(mu_x+mu_y)]^2/[2(sigma_x^2+sigma_y^2)]),](/images/equations/NormalSumDistribution/NumberedEquation1.svg) |
(1)
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which has mean
 |
(2)
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and variance
 |
(3)
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By induction, analogous results hold for the sum of 
normally distributed
variates.
An alternate derivation proceeds by noting that
 |  | ![F_t^(-1){[phi(t)]^n}(x)](/images/equations/NormalSumDistribution/Inline8.svg) |
(4)
|
 |  |  |
(5)
|
where 
is the characteristic function and ](/images/equations/NormalSumDistribution/Inline13.svg)
is the inverse Fourier transform, taken with
parameters 
.
More generally, if 
is normally distributed
with mean 
and variance 
, then a linear function of 
,
 |
(6)
|
is also normally distributed. The new distribution has mean 
and variance 
, as can be derived using the moment-generating
function
 |  |  |
(7)
|
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(8)
|
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(9)
|
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(10)
|
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(11)
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which is of the standard form with
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(12)
|
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(13)
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For a weighted sum of independent variables
 |
(14)
|
the expectation is given by
 |  |  |
(15)
|
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(16)
|
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(17)
|
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(18)
|
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(19)
|
Setting this equal to
 |
(20)
|
gives
 |  |  |
(21)
|
 |  |  |
(22)
|
Therefore, the mean and variance of the weighted sums of 
random variables are their
weighted sums.
If 
are independent and normally
distributed with mean 0 and variance 
,
define
 |
(23)
|
where 
obeys the orthogonality condition
 |
(24)
|
with 
the Kronecker delta. Then 
are also independent and normally distributed with mean
0 and variance 
.
Cramer showed the converse of this result in 1936, namely that if 
and 
are independent
variates and 
has a normal distribution, then both 
and 
must be normal. This result is known as Cramer's theorem.
