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?
Calculating the variance of the ratio of random variables
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probability
statistics
correlation
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2What makes you say that is the solution? – 2011-05-22
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6$\var(X-Y) = \var(X) + \var(Y) - 2 \var(X) \var(Y) \mathrm{corr}(X,Y)$ and not $\var(X/Y)$ – 2011-05-22
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3Sanity check your formula for $Y=X$, then your formula should vanish which is not the case here. – 2011-05-22
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0Cardinal: It gives the correct answer, and that formula was used in a video feed I saw. I just want to understand how to derive it. – 2011-05-22
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0Listing: If Y = X, we should have var(X) + var(X) - 2var(X)*1 = 0? – 2011-05-22
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0@Tomas: Try plugging in $X=0$ in the formula you have, to realize it is not true – 2011-05-22
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0Thomas no at the right side you have $2 \var(X)^2$ which is in the general case not equal to $2 \var(X)$ – 2011-05-22
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2Don't believe everything which you see in a video feed! – 2011-05-22
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11Strictly $\var(X−Y)=\var(X) + \var(Y) - 2 \sqrt{\var(X) \var(Y)} \mathrm{corr}(X,Y)$. Otherwise there is a dimension problem. – 2011-05-22
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0The note given in the link below explains the topic nicely http://www.stat.cmu.edu/~hseltman/files/ratio.pdf – 2015-06-14