I want to compute the variance of $f(X, Y) = XY$, where $X$ and $Y$ are randomly independent.
If I use the definition for the variance $Var[X] = E[(X-E[X])^2]$ and replace $X$ by $f(X,Y)$ I end up with the following expression
$$Var[XY] = Var[X]Var[Y] + Var[X]E[Y]^2 + Var[Y]E[X]^2$$
I have found this result also on Wikipedia: here
However, I also found this approach, where the resulting formula is
$$Var[XY] = 2E[X]E[Y]COV[X,Y]+ Var[X]E[Y]^2 + Var[Y]E[X]^2$$
where the first term is zero since $X$ and $Y$ are independent.
I would like to know which approach is correct for independent random variables? Or are they actually the same and I miss something?
Best Answer
Due to independence of $X$ and $Y$ and of $X^2$ and $Y^2$ we have
\begin{align} {\rm Var}[XY]&=E[X^2Y^2]-E[XY]^2=E[X^2]\,E[Y^2]-E[X]^2\,E[Y]^2\\ &={\rm Var}[X]\,{\rm Var}[Y]+E[X^2]\,E[Y]^2+E[X]^2\,E[Y^2]-2E[X]^2E[Y]^2\\ \tag{1} &={\rm Var}[X]\,{\rm Var}[Y]+{\rm Var}[X]\,E[Y]^2+{\rm Var}[Y]\,E[X]^2\,. \end{align} This is your first formula.
As far as I can tell the authors of that link that leads to the second formula are making a number of silent but crucial assumptions:
First, they assume that $X_i-\overline{X}$ and $Y_i-\overline{Y}$ are small so that approximately $$\tag{2} X_iY_i-\overline{XY}\approx(X_i-\overline{X})\overline{Y}+(Y_i-\overline{Y})\overline{X}\, $$ holds. If we knew $\overline{XY}=\overline{X}\,\overline{Y}$ (which is not necessarly true) formula (2) (which is their (10.7) in a cleaner notation) could be viewed as a Taylor expansion to first order. The authors write (2) as an equation and stay silent about the assumptions leading to it. By squaring (2) and summing up they obtain $$\tag{10.13*} \sigma_{XY}^2\approx \sigma_X^2\overline{Y}^2+\sigma_Y^2\overline{X}^2+2\,{\rm Cov}[X,Y]\overline{X}\,\overline{Y}\,. $$ This is in my opinion an cleaner notation of their (10.13).
The whole story can probably be reconciled as follows: If $X$ and $Y$ are independent then $\overline{XY}=\overline{X}\,\overline{Y}$ holds and (10.13*) becomes $$\tag{3} \sigma_{XY}^2\approx \sigma_X^2\overline{Y}^2+\sigma_Y^2\overline{X}^2\,. $$ The assumption that $X_i-\overline{X}$ and $Y_i-\overline{Y}$ are small is not far from assuming ${\rm Var}[X]{\rm Var}[Y]$ being very small. Hence your first equation (1) approximately says the same as (3).
BTW, the exact version of (2) is obviously $$ \tag{4} X_iY_i-\overline{X}\,\overline{Y}=(X_i-\overline{X})\overline{Y}+(Y_i-\overline{Y})\overline{X}+(X_i-\overline{X})(Y_i-\overline{Y})\,. $$ and this holds without the assumpton that $X_i-\overline{X}$ and $Y_i-\overline{Y}$ are small. Not sure though if a useful equation for $\sigma^2_{XY}$ can be derived from this.