Finding Cov(4X,4X+2Y)

Question

Let X and Y be stochastic variables. Assume Var(X)=1.74 and Cov(X,Y)=0.79

What is Cov(4X,4X+2Y). 

Have tried to use the formulas for Var and Cov to get around this problem, but haven't managed to find a proper way. Any help appreciated. 

Answer 1

There is a property of covariance called bilinearity which is that the covariance of a linear combination.

In general, for constants a,b,c,d and random variables W,X,Y,Z:

Cov(aX+bY, cW+dZ) = acCov(X,W) + adCov(X,Z) + bcCov(Y,W)+ bdCov(Y,Z)

Use the above formula to find Cov(4X,4X+2Y).

Note a=4, b=0, c=4, d=2, W=X, Z=Y

Cov(4X,4X+2Y)= 4*4Cov(X,X) + 4*2Cov(X,Y) + 0*2Cov(Y,X)+ 0*2Cov(Y,Y)

= 16*Var(X) + 8*Cov(X,Y)

Then use the given numbers to calculate it.