True or False: If and are random variables, then

Question

True or False: If  and  are random variables, then

BlackTom AI · Accepted Answer

Consider what Cov(X, Y) means in probability theory and how it is computed. 
Option 1 (True): Cov(X, Y) is defined as E[(X − E[X])(Y − E[Y])]. Expanding this product gives E[XY − X E[Y] − E[X] Y + E[X] E[Y]]. By linearity of expectation, this becomes E[XY] − E[X] E[Y] − E[X] E[Y] + E[X] E[Y] = E[XY] − E[X] E[Y]. Therefore Cov(X, Y) = E[XY] − E[X] E[Y], so the statement is correct. 
Option 2 (False): If someone claimed Cov(X, Y) equals something else, like E[X] + E[Y] or E[XY] alone, that would ignore the cross-term structure in the covariance definition and the subtraction of the product of the means. This misinterpretation fails to account for how deviations from the means interact, which is essential to covariance. 
Summary note: The standard identity Cov(X, Y) = E[XY] − E[X] E[Y] is a direct result of expanding the definition of covariance and applying linearity of expectation, and it does not reduce to E[XY] alone or to a sum of expectations without the product of the means.

AMS 310 Homework 7

True or False: If and are random variables, then

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