15.4: Linear Combinations of Independent Gaussian Random Variables

Let X₁, X₂, . . ., X_n be n independent random variables with means µ₁, µ₂, . . ., µ_n and variances σ²₁ , σ²₂ , . . ., and σ²_n . Let Y be a random variable that is a linear combination of X_i with weights a_i 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

\[\sigma _{Y}^{2}=\sum_{i=1}^{n}a_{i}^{2}\sigma _{i}^{2}\]

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