True or False: , where and .

Question

True or False: , where  and .

BlackTom AI · Accepted Answer

Consider what the equation is saying in terms of basic expectations.
Option analysis start with the identity: E[(X-μx)(Y-μy)] expands to E[XY - Xμy - Yμx + μxμy].
If we separate expectations and use μx = E[X] and μy = E[Y], the expression becomes E[XY] - μyE[X] - μxE[Y] + μxμy, which simplifies to E[XY] - μxμy because E[X] = μx and E[Y] = μy.
Therefore, E[XY] - μxμy = E[(X-μx)(Y-μy)].
Now compare this to the statement in the question: E(XY) − μxμy = E[(X − μx)(Y − μy)], which matches the derived identity.
Option True: This is exactly the equality that follows from expanding the product (X − μx)(Y − μy) and taking expectations.
Option False: If one believed the equality did not hold, a misconception would be that cross-terms cannot be simplified or that the product expansion under expectation yields something different; however, the cross-terms E[X] and E[Y] correctly cancel to give μxμy, not something else.
Another potential trap would be thinking covariance is defined incorrectly; remember Cov(X,Y) = E[(X − μx)(Y − μy)] and also Cov(X,Y) = E[XY] − μxμy, which are the same quantity expressed in two forms.
Thus, the statement is true because it is simply the standard identity for E[XY] in terms of centered variables and their means.

AMS 310 Homework 7

True or False: , where and .

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