Covariance With a Sum

Statistics

Covariance

Medium

X

and

Y

are independent with

Var (X) = 4

and

Var (Y) = 5

. Find

Cov (X, X + Y)

Hint 1

Expand $Cov (X, X + Y)$ using bilinearity of covariance.

Hint 2

Use independence to eliminate the cross term.

Show worked solution

1. Bilinearity
$Cov (X, X + Y) = Cov (X, X) + Cov (X, Y) = Var (X) + Cov (X, Y)$ .
2. Independence kills the cross term
Since $X ⊥ Y$ , $Cov (X, Y) = 0$ , leaving $Cov (X, X + Y) = Var (X) = 4$ .

Answer: 4

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1. Bilinearity