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When we have a ratio of random variables, is their expectation/variance defined in the same way? That is, if we want to write out explicitly $E[\frac{X}{Y}]$ where X and Y are random variables, then

$$E[\frac{X}{Y}] = \int_\Omega \frac{X}{Y} p(x,y)dxdy$$

And similarly, is the variance of the ratio also defined the same way: $$V[\frac{X}{Y}] = \int_\Omega [\frac{X}{Y}-\mu_{\frac{x}{y}}]^2 p(x,y)dxdy$$

for some $\mu_{\frac{x}{y}}$?

This is a definitional question, I ask because I have not seen anyone write out the expectation of ratio of random variables explicitly.

information_interchange
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  • Seems like you might need to use an approximation? https://math.stackexchange.com/questions/40713/calculating-the-variance-of-the-ratio-of-random-variables and https://www.stat.cmu.edu/~hseltman/files/ratio.pdf – information_interchange Feb 13 '20 at 02:48
  • 1. You should not have uppercase letters inside the integrals; in each case you should have $x/y$ where you have $X/Y$. 2. Since your posted questions ask about what something *represents* (and the title asks about notation) it's difficult to see how your comment is relevant to that issue. – Glen_b Feb 13 '20 at 04:15
  • Good point about the uppercase inside the integral, I agree it's confusing. For the second point, I think after searching, I have found that in general, there is no closed form for representing the expectation of the ratio, hence why you $\{must}$ use the approximation – information_interchange Feb 13 '20 at 04:22
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    As for *representing* the expectation of the ratio, you have done so in your question (aside from the error I mentioned). However, *evaluating* it is a different question. Did you instead mean to ask about that? – Glen_b Feb 13 '20 at 04:40
  • Could you explain what you mean by "represent"? – whuber Feb 13 '20 at 15:40
  • See https://stats.stackexchange.com/search?tab=votes&q=LOTUS. – whuber Feb 14 '20 at 18:48
  • Thanks, I edited the question to be more clear. – information_interchange Feb 14 '20 at 18:48
  • Thanks, for the link to the other question. I believe it will answer my question. But to answer my own question, yes I believe that what I've written is the correct explicit definition of the expectation of the ratio of RVs – information_interchange Feb 14 '20 at 18:51
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    It's unclear what your expression means, because it mixes elements of the abstract theoretical integral (over $\Omega$) with the integral over the distribution. For a continuous bivariate distribution with density $p,$ for instance, the LOTUS tells you that $$E\left[\frac{X}{Y}\right]=\iint_{\mathbb{R}^2} \frac{x}{y}\,p(x,y)\,\mathrm{d}x\mathrm{d}y.$$ – whuber Feb 14 '20 at 21:06
  • Thanks @whuber I am sorry for using this confusing notation. I am trying to re-learn a lot of probability/statistics from a more rigorous viewpoint than what I was originally taught. I guess it doesn't work that way where we can be "half-rigorous" – information_interchange Feb 14 '20 at 21:39
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    Probability and statistics can be an extremely confusing subject. I have learned that when I'm unsure about the mathematical notation for something, there's probably a gap in my understanding so I go back to review it. That has been a helpful discipline. – whuber Feb 14 '20 at 21:43

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