Introduction to Probability

CHAPTER 1TWO‐DIMENSIONAL DISCRETE RANDOM VARIABLES AND DISTRIBUTIONS

Francis Galton (Birmingham 1822 – Haslemere, England 1911)

Sir Francis Galton, a half‐cousin of Charles Darwin, was born on 16 February 1822, in Birmingham, England, and died on 17 January 1911, at the age of 88, in Haslemere, England. He was a renowned statistician, sociologist, psychologist, and anthropologist, in addition to being a tropical explorer and geographer. Galton published numerous papers and books. He was very much interested in the joint behavior of quantities, created the statistical concept of correlation, and promoted regression toward the mean, a method which is currently of widespread use in all areas of statistical applications.

1.1 INTRODUCTION

Quite often, in order to study a random experiment, it is not enough to observe a single characteristic (i.e. a random variable); we might be interested in the study of two or more (usually numerical) characteristics. For example:

• when rolling two dice, we focus on both the indication of the first and the second die;
• when studying the operation of a gas station it makes sense to look at the number of cars waiting to be served in each of the gas pumps of the station.

In these cases, apart from the study of each random variable separately, the determination of the behavior of one in relation to the behavior of the others might be of particular ...