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for example, x is random variable, f(x) and g(x) are two "very very" different distribution functions. It is impossible to calculate variance analytically, so if we want to compare var(f) and var(g), what are the potentially possible methods?

maybe, sometimes, we know some other information, such as convexity,slope of the function f and g, etc. what we can do?

are there some applied probability papers? I know this is a very simple question:)

thanks a lot.

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This problem is underspecified. You haven't told us anything about the distributions except for the properties they <i>don't</i> have, i.e. that they are different and that they don't have analytic variances. At this point, we don't know that they have variances at all, or even means, or are even computable. What you need to do is to tell us everything you <i>do</i> know about the distributions. For example, do they have compact support? Is there some way to sample from the distributions? What space is the random variable over? –  Carl Feynman May 21 '13 at 0:29
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