Need help deriving an equation for Covariance?

Question

I need help deriving the following equation for covariance:

cov(X1,X2)=E((X1-mu1)(X2-mu2))

I think I can use the equation cov(X1,X2)=E(X1X2)-E(X1)E(X2) in the derivation if needed.

Answer 1

Are you sure you're supposed to derive cov(X1,X2)=E[(X1-mu1)(X2-mu2)]? That's the definition of correlation in my text. Perhaps you've been asked to show that this equals E[X1*X2] - mu1*mu2? (Which is the same as E[X1X2]-E[X1]E[X2], since E[X1] = mu1 and E[X2] = mu2)

If cov(X1,X2)=E[(X1-mu1)(X2-mu2)] is also the definition in your text, just multiply out (X1-mu1)(X2-mu2) and expand the expectation of the result:

E[(X1-mu1)(X2-mu2)] = E[X1*X2 - mu1*X2 - mu2*X1 + mu1*mu2]
= E[X1*X2] - E[mu1*X2] - E[mu2*X1] + E[mu1*mu2]
= E[X1*X2] - mu1*E[X2] - mu2*E[X1] + mu1*mu2
= E[X1*X2] - mu1*mu2 - mu2*mu1 + mu1*mu2
= E[X1*X2] - mu1*mu2

You could do these steps in reverse order if you truly need to start with E[X1*X2] - mu1*mu2 and end up with E[(X1-mu1)(X2-mu2)]