The danger here is, of course, the side-condition of independence, which can make inhabitants of that type very difficult to reason about. e.g. x + x and 2*x in that world are very different.
Yes. I was surprised (maybe i shouldn't have been): sampler = do x <- gauss 0.0 1.0 y <- gauss 0.0 1.0 return $ (2*x, x+y) main = do xys <- take 100000 `fmap` runSamplerIO sampler print $ runStat (both (before varF fst) (before varF snd)) $ xys => (3.9988971177326498,2.0112123117664975) that is, (variance of 2*x, variance of x+y) The problem is that when you say x+x you don't really mean it; you mean something like liftM2(+) xdist xdist in a probability monad. Had I changed "x+y" to "x+x", I would obviously have gotten identical variances. So maybe referential transparency is not lost after all. Tom