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Let X and Y be independent random variables with $$E(X) = 0\ and\ Y > 0$$ Find the mean value of $$ X/Y$$

My attempt: We have for independent random variables $$E(XY)=E(X)\times E(Y)$$ Hence, $$E(X/Y)=E(X)\times E(1/Y)=0$$ since $$E(X)=0$$

Is this a valid result?

Sina Ahh
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  • Yes, if they are independent, then yes – Fr1 Aug 29 '19 at 18:11
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    @Fr1 Not necessarily. For instance, when $X$ is normal and $Y$ is half-normal, the ratio is Cauchy *which does not have an expectation.* See https://stats.stackexchange.com/questions/299722. – whuber Aug 29 '19 at 18:17
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    To slightly contradict the comment by @Fr1, the ciaim is correct _provided_ $E\big[\frac 1Y\big]$ is finite. Not all positive random variables enjoy this property. – Dilip Sarwate Aug 29 '19 at 18:20
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    @DilipSarwate provided that it is finite, correct, I saw that this week on math stackexchange as I was trying to invert one.. clearly if it is not finite then you would have something like inf*0, which is still undetermined, correct. Thanks.. – Fr1 Aug 29 '19 at 18:26
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    @whuber I said undetermined, but is it underdetermined, or overdetermined? :-) I am joking.. it reminded me the early afternoon, I was about to write “underdetermined” instead of “undetermined”:-) – Fr1 Aug 29 '19 at 18:40
  • I like Dilip Sarwate's the best. I think it is the clearest & easiest to understand. – Michael R. Chernick Aug 29 '19 at 18:52
  • We see an illustration here why I posted a meta question some years ago [*A difficulty with self-study-like questions of the form “Is this correct?”*](https://stats.meta.stackexchange.com/q/2685/805) (though in this case, the answer isn't quite correct, as noted). People are happy to post comments saying "yes" or "no, not quite", but unless the question has an issue requiring more than a few words of explanation, nobody posts an answer. Such an outcome often makes me wonder if such questions are really suitable for this site at all. – Glen_b Aug 31 '19 at 04:58

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