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What is the formula for variance of product of dependent variables? In the case of independent variables the formula is simple: $$ {\rm var}(XY) = E(X^{2}Y^{2}) - E(XY)^{2} = {\rm var}(X){\rm var}(Y) + {\rm var}(X)E(Y)^2 + {\rm var}(Y)E(X)^2 $$ But what is the formula for correlated variables? By the way, how can I find the correlation based on the statistical data? |
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Well, using the familiar identity you pointed out, $$ {\rm var}(XY) = E(X^{2}Y^{2}) - E(XY)^{2} $$ Using the analogous formula for covariance, $$ E(X^{2}Y^{2}) = {\rm cov}(X^{2}, Y^{2}) + E(X^2)E(Y^2) $$ and $$ E(XY)^{2} = [ {\rm cov}(X,Y) + E(X)E(Y) ]^{2} $$ which implies that, in general, ${\rm var}(XY)$ can be written as $$ {\rm cov}(X^{2}, Y^{2}) + [{\rm var}(X) + E(X)^2] \cdot[{\rm var}(Y) + E(Y)^2] - [ {\rm cov}(X,Y) + E(X)E(Y) ]^{2} $$ Note that in the independence case, ${\rm cov}(X^2,Y^2) = {\rm cov}(X,Y) = 0$ and this reduces to $$ [{\rm var}(X) + E(X)^2] \cdot[{\rm var}(Y) + E(Y)^2] - [ E(X)E(Y) ]^{2} $$ and the two $[ E(X)E(Y) ]^{2}$ terms cancel out and you get $$ {\rm var}(X){\rm var}(Y) + {\rm var}(X)E(Y)^{2} + {\rm var}(Y)E(X)^{2} $$ as you pointed out above. Edit: If all you observe is $XY$ and not $X$ and $Y$ separately, then I don't think there is a way for you to estimate ${\rm cov}(X,Y)$ or ${\rm cov}(X^2,Y^2)$ except in special cases (for example, if $X,Y$ have means that are known a priori) |
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