# 4MULTIPLE RANDOM VARIABLES

## 4.1 INTRODUCTION

In many experiments an observation is expressible, not as a single numerical quantity, but as a family of several separate numerical quantities. Thus, for example, if a pair of distinguishable dice is tossed, the outcome is a pair (*x, y*), where *x* denotes the face value on the first die, and *y*, the face value on the second die. Similarly, to record the height and weight of every person in a certain community we need a pair (*x, y*), where the components represent, respectively, the height and weight of a particular individual. To be able to describe such experiments mathematically we must study the *multidimensional random variables*.

In Section 4.2 we introduce the basic notations involved and study joint, marginal, and conditional distributions. In Section 4.3 we examine independent random variables and investigate some consequences of independence. Section 4.4 deals with functions of several random variables and their induced distributions. Section 4.5 considers moments, covariance, and correlation, and in Section 4.6 we study conditional expectation. The last section deals with ordered observations.

## 4.2 MULTIPLE RANDOM VARIABLES

In this section we study multidimensional RVs. Let (Ω, , *P*) be a fixed but otherwise arbitrary probability space.

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