# Variance of product of dependent variables

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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why do you put [var(X)+E(X)2]⋅[var(Y)+E(Y)2] instead of E(X2)E(Y2)??? –  ewe Nov 28 at 19:53