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I am a little stuck with my project. In the calculations of my project, I need to calculate the spread of some random variables. Up to now, there was no special difficulty to analytically calculate the standard deviation of correlated and uncorrelated random variables.

The point is that I am dealing now with variances and covariances of ratios between 3 different random variables X, W and Y. How to calculate the following? $$ cov(X/Y, W/Y) = \,? $$ Since the ratios share the same denominator, is it correct to write : $$ cov(X/Y, W/Y) = \left[\frac{1}{E(Y)}\right]^2\,cov(X, W)\,? $$

kjetil b halvorsen
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ehdu
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1 Answers1

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The formula you have given is not correct. You need the law of total covariance, given in Covariance of a compound distribution

Using that we can calculate $$ \DeclareMathOperator{\cov}{Cov} \DeclareMathOperator{\E}{E} \cov(\frac{X}{Y}, \frac{W}{Y}) = \E \cov(\frac{X}{Y}, \frac{W}{Y} \mid Y) + \cov(\E(\frac{X}{Y} \mid Y), \E( \frac{W}{Y} \mid Y)) \\ = \E ( (\frac1{Y})^2 \cov(X,W\mid Y)) + \cov(\frac1{Y}\E(X\mid Y), \frac1{Y}\E(W\mid Y) ) $$ Without more information (than you have given) about the joint distribution of $X,Y,W$ we cannot simplify more. In your guessed formula you have totally lost the second term, but the first term here isn't equal to your guessed formula either. Specifically, even if $\cov(X,W \mid Y)$ do not depend on $Y$, the first term would be $\cov(X,W) \E ( (\frac1{Y})^2 )$ and that last expectation of a ratio cannot be simplified further (without more information). See I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

kjetil b halvorsen
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