Symmetry of a ratio of two random variables My question is whether the fact that two random variables, X and Y, are symmetrically distributed implies that their ratio, Z=X/Y, is symmetrically distributed too. 
 A: If values of $X$ and $Y$ are allowed to be negative, zero or positive, then quite a range of behaviour is possible, some perverse. Whether the variables are independent or correlated is crucial. 
It is better to say what else is true, for cases you care about. 
Although there are exceptions, ratios make sense mostly when both components are positive, or at least the denominator. Characteristically ratios are then (possibly highly) skewed in distribution, but their logarithms behave better. Hand waving underlines that when both values are positive, $X < Y$ implies a ratio between 0 and 1, $X = Y$ a ratio of 1 and $X > Y$ a ratio greater than 1 without definite upper limit, so in combination skewness is likely. 
The easiest way to explore this is to simulate from plausible distributions for each variable, either independently (very easy) or with a stated correlation (may be trickier). 
A: No, not in general. 
Consider X,Y both uniform on (0,1). Their ratio is not symmetric; it's heavily right skew. [However in this particular example the log of their ratio is symmetric.]
What examples did you try? Note that even if you can't do the algebra, simulation is a straightforward way to find counterexamples. Here's a histogram of the ratio of two standard uniforms:

The upper tail is much longer (the largest ratio in my sample exceeds 8000); I cut the upper tail off (about 1.6% of the sample) so you could see some detail of the shape of the main part of the distribution.
The following R code will generate a plot like that:
ratio=runif(10000)/runif(10000)
hist(ratio[ratio<30],n=300,freq=FALSE)

A: If by symmetrically distributed you mean that their distributions are symmetric around zero, then the answer is no in general, unless they are independent. For instance, take $X=Y$.
