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What is the standard deviation of the sum of two variables?
Standard Deviation of the Sum/Difference of Two Independent Random Variables. Sum: For any two independent random variables X and Y, if S = X + Y, the variance of S is SD^2= (X+Y)^2 . To find the standard deviation, take the square root of the variance formula: SD = sqrt(SDX^2 + SDY^2).
What is the variance of a sum?
For independent random variables X and Y, the variance of their sum or difference is the sum of their variances: Variances are added for both the sum and difference of two independent random variables because the variation in each variable contributes to the variation in each case.
When does the variance of a sum equal the sum of the variances?
So, if the covariances average to 0, which would be a consequence if the variables are pairwise uncorrelated or if they are independent, then the variance of the sum is the sum of the variances. An example where this is not true: Let Var ( X 1) = 1. Let X 2 = X 1. Then Var ( X 1 + X 2) = Var ( 2 X 1) = 4.
When does the sum of the covariances average to 0?
So, if the covariances average to 0, which would be a consequence if the variables are pairwise uncorrelated or if they are independent, then the variance of the sum is the sum of the variances. An example where this is not true: Let Var (X1) = 1. Let X2 = X1. Then Var (X1 + X2) = Var (2X1) = 4.
What is the theorem about variance and covariance?
Fact:If two RV’s are independent, we can’t predict one using the other, so there is no linear association, and their covariance is 0. Theorem:If X and Y are independent, then Var(X +Y) = Var(X)+Var(Y) In other words, if the random variables are independent, then the variance of their sum is the sum of their variances!
How to find the sum of random variables?
However, if X, Y are independent, then E ( X Y) = E ( X) E ( Y), and we have Var ( X + Y) = Var ( X) + Var ( Y). Notice that we can produce the result for the sum of n random variables by a simple induction.