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I want to calculate $\newcommand{\var}{\mathrm{var}}\var(X/Y)$. I know that the solution is $\var(X) + \var(Y) - 2 \var(X) \var(Y) \mathrm{corr}(X,Y) \>,$ but, how do I derive it from "common" rules of variance calculations?

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    The note given in the link below explains the topic nicely http://www.stat.cmu.edu/~hseltman/files/ratio.pdf2015-06-14

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As others have noted, the formula you provide is incorrect. For general distributions, there is no closed formula for the variance or a ratio. However, you can approximate it by using some Taylor series expansions. The results are presented (in strange notation) in the this pdf file.

You can work it out exactly yourself by constructing the Taylor series expansion of $f(X,Y)=X/Y$ about the expected $X$ and $Y$ (E$[X]$ and E$[Y]$). Then look at E$[f(X,Y)]$ and E$[\left(f(X,Y)-\right.$E$\left.[f(X,Y)]\right)^2]$ using those approximations.

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    A useful form of the result you linked to, expressed in terms of expected values, is $ \operatorname{Var}\frac CD\approx\frac{\langle CC\rangle\langle D\rangle\langle D\rangle-2\langle CD\rangle\langle C\rangle\langle D\rangle+\langle DD\rangle\langle C\rangle\langle C\rangle}{\langle D\rangle^4}\;. $2013-01-19
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The ratio of two random variables does not in general have a well-defined variance, even when the numerator and denominator do. A simple example is the Cauchy distribution which is the ratio of two independent normal random variables. As Sivaram has pointed out in the comments, the formula you have given (with the correction noted by Henry) is for the variance of the difference of two random variables.

Henry's wiki link includes a formula for approximating the ratio distribution with a normal distribution. It seems like this can be used to create the sort of formula you are looking for.

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    And to nitpick even more, the standard Cauchy distribution is the ratio of two independent _zero-mean_ normal random variables with _identical variance_. They don't have to be _standard_ normal random variables as long as they are zero-mean (cf. @Henry's comment) and have equal variance. Also, the standard Cauchy random variable is the ratio of any two random variables whose joint density has _circular symmetry_ about the origin. Such random variables are dependent except when they are marginally normal.2012-04-18