BabakP's answer is a good one. Read it, but I'd like to add a few points.
The computation of the distribution of a transformation of a random variable (or of several variables) is often called statistical tolerancing. A problem is, for most such transformations, there is no simple distribution that works. Thus suppose X is a Gaussian random variable, then F(X), where F is a linear function, also has a Gaussian distribution, but with an adjusted mean and variance, based on the transformation L, and the mean and variance of X.
However, things get nasty almost always when F(X) is at all nonlinear. Ok, if F is the exponential function, then F(X) for normal X has a lognormal distribution. But most nonlinear functions will not give you some well known distribution. So what can you do?
One simple solution is to compute the mean and variance of the transformation. Given a mean and variance, one might choose to assume a normal distribution with that mean and variance. One might even do more, computing the first four moments of F(X). There are several families of distributions (Pearson & Johnson families) that allow you to find a distribution that matches the moments you have just found.
So, the question is, how might one compute those moments? The simple answer is if you knew the derivative of F, then approximating F by a truncated (first order) Taylor series around the mean of X can allow you to find approximations for those moments. Essentially, if F is well approximated by a linear function over the support of X, then those moments will be good estimates. (For a Gaussian distribution, the support might be considered to be something like +/-6 sigma.)
Others have used second order Taylor series approximations. Here too we can compute approximate moments of the transformation.
And finally, I might mention Taguchi methods, as well as modified Taguchi methods. They too allow you to find approximations of the moments. A nice thing about the modified Taguchi methods is they are based on Gaussian integrations, and allow you to use higher order approximations quite easily, without any need to compute the derivatives of your transformation F.
Another nice feature of the modified Taguchi methods is they easily allow you to formulate a method that works on uniform random variables X, or gamma random variables, etc. In fact, there are schemes that will allow you to solve for the moments of a wide variety of random variables.