Question

If X and Y are two random variables and a and b are constants, which of the following is true about the equation Cov(X, aY+bZ)=a.Cov(X,Y)+b.Cov(X,Z)? A) The equation is only true if X, Y, and Z are independent. B) The equation is always true regardless of the dependence between X, Y, and Z. C) The equation is only true if X, Y, and Z are identically distributed. D) The equation is only true if X, Y, and Z have the same variance

Answer 1

The equation Cov(X, aY+bZ)=a.Cov(X,Y)+b.Cov(X,Z) is true in all cases for any constants a and b, and random variables X, Y, and Z, due to the properties of covariance being a linear operator.

Step-by-step explanation:

The question you've asked pertains to the properties of covariance in statistics and whether the equation Cov(X, aY+bZ)=a.Cov(X,Y)+b.Cov(X,Z) is true based on the dependence or independence of the variables X, Y, and Z. The correct answer to this question is B) The equation is always true regardless of the dependence between X, Y, and Z. This is because covariance is a linear operator with respect to scalar multiplication and addition of random variables. Essentially, for any constants a and b, and random variables X, Y, and Z, the following expansion is true due to the bilinearity property of covariance:

Cov(X, aY + bZ) = a Cov(X, Y) + b Cov(X, Z)

This property holds without any requirement for independence or identical distribution of the random variables involved.