CS(13) Transformation Methods: Sums of Random Variables

This topic focuses on Transformation Methods and Sums of Random Variables, which expand the tools for generating random variables to include approaches beyond the inverse transformation method. these methods are particularly useful for distributions where the cumulative distribution function (CDF) is not easily invertible or when the desired distribution arises from combining or modifying simpler distributions.

Transformation methods rely on the mathematical properties of random variables to create new distributions from existing ones. The concept of “sums of random variables” explores how the sum of independent variables can form a new random variable, whose distribution is either known or derivable. Let’s examine this in detail.

Transformation Methods

The essence of transformation methods is to apply a deterministic mathematical operation to one or more random variables to generate a variable with a specific distribution. A well-known example is the generation of Standard Normal random variables Z ∼ N(0, 1) from uniform random variables U ∼ U(0, 1). This can be achieved using the Box-Muller transform method that maps two uniform random variables into two independent standard normal variables. The Box-Muller transformation used the following equations: