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Say I have two Gaussian distributions. Are the following two questions equivalent?

[a] What is the ratio of their averages?

[b] What is the average ratio of two random samples drawn, one from each distribution?

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The answer to [a] is a number while the answer to [b] is a (nondegenerate) random variable hence these cannot coincide. What happens under suitable hypotheses however, is that the random variable may converge almost surely to the number when the size of the sample goes to infinity. – Did Jan 22 '12 at 14:36
@DidierPiau: To me it reads like the answer to [b] is a number as well, namely the expectation of $X/Y$ where $X$ and $Y$ are (independent?) Gaussian variables. – Rasmus Jan 22 '12 at 15:14
@Rasmus: You are right, I missed the word average in [b]... Sorry about that. Unfortunately, this average does not exist (at least when X and Y are independent and when the Gaussian distribution used for Y is not a nonzero Dirac). – Did Jan 22 '12 at 15:38
@DidierPiau: It is not clear to me why "the average does not exist"... Isn't the distribution of X/Y well-defined? Thanks. – Omri Jan 22 '12 at 16:04
Omri: Of course the distribution is well defined (??) but it is not integrable. – Did Jan 22 '12 at 16:06

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