Whe know that $\mathrm{Cov}(X,Y)=E(X.Y)-E(X).E(Y)$ so if we consider this equation what happened if $X$ & $Y$ being conditional random variable $\mathrm{Cov}((X|Z_i),(Y|Z_j))=?$
Expansion Conditional Variance
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statistics
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2OK, I didn't see that, but your notation makes no sense. There isn't such a thing as a conditional random variable. What you get is $\text{Cov}(X,Y \mid Z_i, Z_j) = E[XY\mid Z_i, Z_j] - E[X\mid Z_i, Z_j]E[Y\mid Z_i, Z_j].$ – 2012-11-01