Suppose X and Y are two random variables such that rho(X,Y)=1/2, Var x=1, and Var Y=2. Compute Var (X-2Y)

Alberto Calhoun

Alberto Calhoun

Answered question

2022-11-05

Suppose X and Y are two random variables such that ρ ( X , Y ) = 1 / 2, Var X = 1, and Var Y = 2. Compute Var (X - 2Y).

Answer & Explanation

Berattirna

Berattirna

Beginner2022-11-06Added 19 answers

Observe following calculation
Var(X-2Y)=Cov(X-2Y,X-2Y)
=Cov(X,X)-4Cov(X,Y)+4Cov(Y,Y)=Var(X)+4Var(Y)-4Cov(X,Y)
We know that Var(X)=1 and Var(Y)=2 and we can calculate covariance as following
C o v ( X , Y ) = ρ ( X , Y ) V a r ( X ) V a r ( Y ) = 2 2
Therefore
V a r ( X 2 Y ) = 1 + 8 2 2 = 9 2 2
Result:
V a r ( X 2 Y ) = 9 2 2

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