Calculating the variance of the ratio of random variables

I intend to compute $\newcommand{\var}{\mathrm{var}}\var(X/Y)$. I recognize that the remedy is $$\var(X) + \var(Y) - 2 \var(X) \var(Y) \mathrm{corr}(X,Y) \>,$$ yet, just how do I acquire it from "common" regulations of difference estimations?

2022-06-07 14:33:32
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Answers: 2

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

2022-07-22 11:37:54

The proportion of 2 arbitrary variables does not as a whole have a well - specified difference, also when the numerator and also do. A straightforward instance is the Cauchy distribution which is the proportion of 2 independent regular arbitrary variables. As Sivaram has actually mentioned in the remarks, the formula you have actually offered (with the improvement kept in mind by Henry) is for the difference of the distinction of 2 arbitrary variables.

Henry is wiki link consists of a formula for approximating the ratio distribution with a normal distribution. It feels like this can be made use of to create the type of formula you are seeking.

2022-06-07 15:04:19