In a process of proof ; unbiased estimator of the covariance

Question

The link below is proving that sample covariance is an unbiased estimator of the covariance

unbiased estimate of the covariance

But, I still don't understand one thing in one of the answers, which was written by 'Sandipan Dey'

In a fourth line, He mentioned that,

$∑E[X_iY_j]$ (when $i≠j$) $=$ $n(n-1)μ_Xμ_Y$ (Since $X_i$ and $Y_j$ are independent for $i≠j$)

Could you explain in detail that how $X_i$ and $Y_j$ can be independent when $i≠j$ ?

Answer 1

The premise of the question you linked to states that $(X_1,Y_1),\ldots,(X_n,Y_n)$ is an independent sample. Thus $(X_i,Y_i)\perp(X_j,Y_j)$ for $i\ne j$. It follows that $X_i\perp Y_j$ for $i\ne j$.