15.4: Linear Combinations of Independent Gaussian Random Variables
- Page ID
- 14868
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Let X1, X2, . . ., Xn be n independent random variables with means µ1, µ2, . . ., µn and variances σ21 , σ22 , . . ., and σ2n . Let Y be a random variable that is a linear combination of Xi with weights ai so that Y = 
.
As the sum of two Gaussian random variables is again a Gaussian, Y is Gaussian distributed with a mean
\[\mu _{Y}=\sum_{i=1}^{n}a_{i}\mu _{i}\]
and a variance
\[\sigma _{Y}^{2}=\sum_{i=1}^{n}a_{i}^{2}\sigma _{i}^{2}\]