# Covariance simplification

We all know that the covariance between two variables is defined as:

$Cov(X_{i},Y_{i}) = E[(X_{i}-\mu_{x})(Y_{i}-\mu_{Y})]$

Now I have seen this "simplification":

$E[(X_{i}-\mu_{x})(Y_{i}-\mu_{Y}) = E[(X_{i}-\mu_{x})Y_{i})]$

I get that you can simplify that to the usual covariance formula $E[(X_{i}-\mu_{x})Y_{i})]=E(X_{i}Y_{i})-E(X_{i})E(Y_{i})$

I wanted to ask you whether this always holds; or asked differently: When does the specification from above not apply? (This might be obvious but I am kind of confused right now, sorry). Thanks in advance!

• It applies always (if covariance is defined). Observe that $(X-\mu)(Y-\nu)=(X-\mu)Y-(X-\mu)\nu$. Take expectation on both sides and on RHS use linearity of expectation. Then the second term on RHS has value $0$ if $\mathbb EX=\mu$. – drhab Jun 3 '17 at 13:01
• Thank you so much @drhab! This confused me way too long (I have seen it being part of countless proofs already), your explanation really helped. – Kuma Jun 3 '17 at 13:17