Means and Variances

This result for the mean of a sum holds whether or not the random variables are independent.

If the random variables

X

and

Y

are independent, then the variance of the sum

X

+

Y

is given by

Var

[

X

+

Y

] = Var[X] + Var[Y] (2.16)

If the random variables X and Y are not independent, then the variance of the sum X + Y is given by

Var[X + Y] = Var[X] + Var[Y] + 2Cov[X, Y] (2.17)

where Cov[X, Y] is the covariance of X and Y, given by Equation (2.12). For any random variables X and Y, and any constants a and b, the mean and

variance of the linear combination aX + bY are given by

E[aX + bY] = aE[X] + bE[Y] (2.18) and

Var[aX + bY] = a2Var[X] + b2Var[Y] + 2abCov[X, Y] (2.19) respectively.