# Why is Covariance defined as…

Why is Covariance defined as:

• $\operatorname{cov}(X,Y) = E[(X-E[X])(Y-E[Y])]$

My book simply states this identity but doesn't explain how it is derived. I know that covariance can also be written as:

• $\operatorname{cov}(X,Y) = E[XY] -E[X]E[Y]$

I know how to derive this identity from the first definition.

• Consider $\mathbb{E}[XY]$ and think of when $\mathbb{E}[XY]=\mathbb{E}[X] \mathbb{E}[Y]$ – Karl Oct 5 '15 at 5:26
• Expectation is a Linear Operator in the sense $E[aX+bY]=aE[X]+bE[Y]$ – Ekaveera Kumar Sharma Oct 5 '15 at 6:51
• @Karl Done. And then what? – Did Oct 5 '15 at 7:37
• @Did I was thinking that the situation motivates the definition of covariance as $\mathbb{E}[XY]=\mathbb{E}[X] \mathbb{E}[Y]$ when $cov(X,Y) =0$ Is this incorrect? – Karl Oct 5 '15 at 7:57
• @Karl More meaningless than incorrect. "the situation" Which situation? "the definition of covariance as 𝔼[XY]=𝔼[X]𝔼[Y]" 𝔼[XY]=𝔼[X]𝔼[Y] is not a definition of covariance. – Did Oct 5 '15 at 8:01