Transforming Gamma random variable

Question

Suppose I want to transform the Gamma variate like this $Y=(X-\text{mean})/SD$, where $X$ is a gamma variate. I know applying this transformation does not necessarily imply that the distribution of $Y$ is standard Normal. Can I use this transformation on say Gamma or any other skewed variables like lognormal, exponential? I need a reference of a good paper or some book reference for this transformation on Gamma, lognormal, or any other skewed variate.

Answer 1

$$\mathbb{P}(Y\leq y) = \mathbb{P}\left(\frac{X-\mu}{\sigma}\leq y\right)=\mathbb{P}(X\leq \mu + \sigma y) \; .$$

Any probability statement you can make about $Y$ can be translated into a statement about $X$. You already know everything you need to know.

Answer 2

I do not know why you want to transform a gamma density in such way. Usually the standardization transformations are done to get a standard density in order to calculate probabilities using tables.

If $X\sim \text{Gamma}[n;\theta]$ (where $\theta$ is the rate parameter) the standardization is

$$Y=2\theta X$$

so that now $Y\sim \chi_{(2n)}^2$, known distribution and free of $\theta$