Normal Sum Distribution
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
www.almerja.com
الجزء والصفحة:
...
12-4-2021
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Normal Sum Distribution
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)]),](https://mathworld.wolfram.com/images/equations/NormalSumDistribution/NumberedEquation1.gif) |
(1)
|
which has mean
 |
(2)
|
and variance
 |
(3)
|
By induction, analogous results hold for the sum of 
normally distributed variates.
An alternate derivation proceeds by noting that
where 
is the characteristic function and ](https://mathworld.wolfram.com/images/equations/NormalSumDistribution/Inline13.gif)
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
which is of the standard form with
For a weighted sum of independent variables
 |
(14)
|
the expectation is given by
Setting this equal to
 |
(20)
|
gives
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.
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