How to combine distributions

If I have two continuous distributions $f(x)$ and $g(x)$, there are several mathematical ways to combine $f$ and $g$ to get new distributions. Which correspond to what statistical interpretation? For example, if I multiply $f$ and $g$ into $fg$, does $fg$ have a statistical meaning? What about $f/g$, $f \circ g$ and $f \star g$ (convolution)? Also, can we do the same with discrete distributions?

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1 Answer

If $f$ and $g$ represent the densities of independent random variables, then the product $f_X(x)g_Y(y)$ is the density of $(x,y)$ under the joint distribution of $X$ and $Y$. Similarly, for two discrete distributions, the product of their mass functions is the mass function of their joint distribution.

As demonstrated here, convolving the two derives the density or mass of their sum, the random variable $Z = X + Y$. (Note that this only holds when $X$ and $Y$ are independent.)

I suppose one could view $\frac{f(x)}{g(x)}$ as the Bayes factor of two candidate models, both with static parameters, after a single observation $x$. That strikes me as of trivial use, but I'm not aware of another interpretation. (Note though that, as Glen points out below, this cannot be interpreted as the density of some other random variable.)

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Thanks! Any use for f(g(x))? –  Frank Jul 6 '14 at 21:59
(i) convolution only works as the density of the sum if they're independent. (ii) f/g won't be a density, though. Frank: f(g(x)) won't be a density in general. There are occasional uses for such a calculation, just as there are in other areas of mathematics - they just don't produce densities. You may be able to scale to a density, but they're usually not very interesting as practical objects. –  Glen_b Jul 6 '14 at 22:11
@Glen_b Ah! Thank you much, I thought I'd written so regarding convolution. Will correct. Frank, sorry, but Glen_b's comment says all I know, and more, about f(g(x)). –  SeanEaster Jul 7 '14 at 17:57
A (trivial) example where $f(g(x))$ gives a density: $f(x)=g(x)=x, 0<x<\sqrt{2}$ (though strictly we have to deal with things more carefully to make the result defined everywhere). An example where it doesn't even scale to a density (this is the usual case): $f$ and $g$ are both standard normal densities. You can get a bit further with cdfs: $F(G(x))$ can be a cdf if the domain of $F$ is the unit interval (e.g. a beta cdf). –  Glen_b Jul 7 '14 at 23:15