Covariance Alternative Formula

Question

Can someone show how this is equivalent? Cov (X,Y) = E[X(Y – μY)] thank you in advance!

Answer 1

The easiest way to remember the formula for the covariance between two variables (X,Y) is "covariance equals the expectation of the product minus the product of the expectations." In equation form, this is expressed as equation 1:

#1: cov(X,Y) = E[X*Y] - [E(X)*E(Y)]

Note: Equation 1 is actually the result of multiplying out the formal definition for covariance, which is:

DEFINITION: cov(X,Y) = E[(X-E[X])(Y-E[Y])]

We'll start with equation 2 and manipulate it to show it equals equation 1:

#2: E[X(Y-μ_y)]

The expectation operator E[.] is linear in parameters, so we can use the distributive property to multiply the expression out.

→E[XY-Xμ_y]

→E[XY]-E[Xμ_y]

μ_y, the mean of Y, is a constant value. The expectation of a constant is itself. So, E[μ_y] = μ_y.

#3: E[XY]-[E[X]μ_y]

Regarding notation, the expressions E(Y) and μ_y are identical. So, equation 3 can be expressed as equation 4:

#4: E[XY] - [E(X)*E(Y)]

Equation 4 is the right-hand side of equation 1, which equals Cov(X,Y). Thus, we have proven that equation 2, E[X(Y-μ_y)], equals Cov(X,Y).