How do you find the variance of the sum of two independent normally distributed random variables, X and Y, if the two variables are correlated? That is, Var(X+Y) = ___ ?

foyerir

foyerir

Answered question

2022-09-11

How do you find the variance of the sum of two independent normally distributed random variables, X and Y, if the two variables are correlated? That is, Var(X+Y) = ___ ?

Answer & Explanation

Grace Moses

Grace Moses

Beginner2022-09-12Added 13 answers

If X and Y are two independent normally distributed variables with parameters ( m X , σ X 2 ) and ( m Y , σ Y 2 ) then the variance of their sum is equal to
σ X 2 + σ Y 2
This result is proven using two facts.
First, the variance of the sum of two independent random variables is equal to the sum of their variances.
Second, the variance of a normally distributed random variable with parameters ( m , σ 2 ) is equal to σ 2
Combining these two facts we get
Var(X+Y)=Var(X)+Var(Y)
= σ X 2 + σ Y 2

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