# 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

## 2 Answers

Here are a few more.

(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.

I think many statistical analysis uses this. For exampl, car insurance, both the probability of occurrence and the reimbursement amount are r.v.) and when they have these r.v., they may have to do some transformation. For example, they know that the prob. of accident is 0.1 and the amt of reimbursement is a r.v. X, than if they want to gain money, they may set the premium to something like E(0.2X). This is of course not realistic as they have more factors to take into account but may give you a taste why they transform R.V.