## Gaussian Distribution Linear Combination of Variates

If is Normally Distributed with Mean and Variance , then a linear function of ,

 (1)

is also Normally Distributed. The new distribution has Mean and Variance , as can be derived using the Moment-Generating Function
 (2)

which is of the standard form with
 (3)

 (4)

For a weighted sum of independent variables
 (5)

the expectation is given by
 (6)

Setting this equal to
 (7)

gives
 (8) (9)

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

 (10)

where obeys the Orthogonality Condition
 (11)

with the Kronecker Delta. Then are also independent and normally distributed with Mean 0 and Variance .