# Transformation of Random Variables.

What is the use case for transforming random variables from 1 to another. I understand the process but where does it finds its use.

-Ravi

(1) $Theory.$ Suppose $X$ is a random variable with mean $E(X) = \mu.$ The variance of $X$ is defined in terms of a transformation, as follows: $V(X) = E(g(X)),$ where $g(X) = (X - \mu)^2.$
(2) $Geometrical\; application.$ You are given the distribution of a random variable $R,$ which is the radius of a randomly drawn circle. What is the distribution of the area of such a randomly drawn circle? That would be the distribution of the random variable $A = g(R) = \pi R^2.$
(3) $Simulation\; study.$ In probability simulation studies, the (pseudo)random generator of a computer program typically gives values $U$ that behave as a random sample from $Unif(0, 1),$ that is, distributed uniformly on the interval $(0, 1).$ Then values of $X = -\log(U)$ (logarithm base $e$) is a random sample from the exponential distribution with unit mean. (Random samples from other distributions can be generated using other transformations of $U$.)
(4) $Applied\; probability\; modeling.$More generally, in applications it is very common to build a model for a messy application using transformations of simpler, well-understood random variables.