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# Chapter 5Bivariate Probability Distributions

## 5.1 Introduction

So far we have studied a single random variable defined on the points of a sample space. Scientific investigations, however, most commonly involve several random variables arising in the course of an investigation. A physicist, for example, may be interested in studying the effects of transmissions in a fiber optic cable when transmission rates and the composition of the cable are varied; sample surveys usually ask several questions of the respondents creating separate random variables for each question; educators studying grade point averages for college students find that these averages are dependent on intelligence, entrance examinations, rank in high school class, as well as many other factors that could be considered. Each of these examples suggests a sample space on which more than one random variable is defined.

While these variables could be considered individually as the univariate variables studied in the previous chapters, studies of the individual random variables will provide no information at all on how the variables behave together. Separate studies then offer no information on how the variables interact or are correlated with each other; this is often crucial information in scientific investigations, since the manner in which the variables act together may indicate the most important factors in explaining the outcome. Because of this, investigations involving only one factor at a time are becoming ...

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