“Gaussian Distribution Linear Combination of Variates”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gaussian Distribution Linear Combination of Variates

释义

Gaussian Distribution Linear Combination of Variates

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

(1)

is also Normally Distributed. The new distribution has Mean

andVariance

, 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

Setting this equal to

(7)

gives

			(8)
			(9)

Therefore, the Mean and Variance of the weighted sums of

Random Variablesare 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 0and Variance

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