Lesson 23: Transformations of Two Random Variables

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Introduction

In this lesson, we consider the situation where we have two random variables and we are interested in the joint distribution of two new random variables which are a transformation of the original one. Such a transformation is called a bivariate transformation. We use a generalization of the change of variables technique which we learned in Lesson 22. We provide examples of random variables whose density functions can be derived through a bivariate transformation.

Objectives

  • To learn how to use the change-of-variable technique to find the probability distribution of \(Y_1 = u_1(X_1, X_2), Y_2 = u_2(X_1, X_2\), a one-to-one transformation of the two random variables \(X_1\) and \(X_2\).