Need a reference or sample problem(interpreting the property of expectation value)

Question

Given the property $E[XY]=\mu_{X} \mu_Y+ \rho \sigma_X \sigma_Y,$

How to interpret this value intuitively? And any example to understand it? How do people come up with that?

Answer 1

I assume you mean $\rho$ is the correlation of $X$ and $Y$, i.e. $\rho=\frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}}$. I'll answer your last question first: $\operatorname{Cov}(X,Y)=\mathbb{E}(XY)-\mathbb{E}(X)\mathbb{E}(Y)$ by definition, so $\rho$ is exactly defined to satisfy your above equation. You don't define covariance for some separate other reason and then suddenly come up with the identity.

As for how intuitive it is: It just explains to you that if $X$ and $Y$ are not independent, they might have some sort of linear interaction which changes the mean of the product.

If $X\sim Ber(\frac{1}{2})$ and $Y=-X$, then it's very easy to check that $\operatorname{Cov}(X,Y)=-\operatorname{Var}(X)=-\frac{1}{4}$ and $XY=Y,$ so $$ -\frac{1}{2}=\mathbb{E}(Y)=\mathbb{E}(XY)\neq \mathbb{E}(X)\mathbb{E}(Y)=-\frac{1}{4} $$ because when $Y=-1$ (the events that contribute to $\mathbb{E}(Y)$), then we also know that $X=1$, as opposed to the case where $X$ and $Y$are independent, where knowing $Y$ tells you nothing about $X$, so that $X$would be $0$half the time, even given the information that $Y=-1$. The interaction between the two variables thus has a very real influence on the expectation of the product.