Q:

Examples of how to transform a random variable with a distribution not Gaussian to a random variable with Gaussian distribution

My doubt is if there exists a method to transform a random variable with not Gaussian distribution in other with a Gaussian distribution.
I only found a random variable with Birhaum Saunders distribution that I can transform in Gaussian.
I would like to obtain other examples.

A:

If you have a continuous random variable $X$ with distribution function $F_X(X)$, that you know what it is, then the random variable
$$Y=\sigma \Phi^{-1}[F_X(X)] + \mu \sim N(\mu,\sigma^2)$$
where $\Phi()^{-1}$ is the inverse standard Normal distribution function. 
Since this is the standard way of generating draws from a Normal Distribution (since $F_X(X)\sim U(0,1)$), I wonder whether this is what you are really asking here.

