Want to see if this covariance formula I'm thinking of is correct?

Question

In my book, in the section about multiple random variables

I am told that the Covariance of random variables $X_1$ and $X_2$ is Cov($X_1,X_2$) = E($X_1X_2)-\mu_1\mu_2$

My question is, is an equivalent form of the above:

Cov($X_1,X_2$) = E($X_1$)E($X_2$) - $\mu_1\mu_2$?

Answer 1

Not always.   That will only occur when $\mathsf E(X_1X_2)=\mathsf E(X_1)\mathsf E(X_2)$, which is the case when $X_1$ and $X_2$ are linearly uncorrelated.

---

Since, $\mu_1=\mathsf E(X_1)$ and $\mu_2=\mathsf E(X_2)$, then $\mathsf E(X_1)\mathsf E(X_2)-\mu_1\mu_2=0$.