Biagini Francesca - Elements of Probability and Statistics: an Introduction to Probability with de Finettis Approach and to Bayesian Statistics

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Part I
Probability

Springer International Publishing Switzerland 2016

Francesca Biagini and Massimo Campanino Elements of Probability and Statistics UNITEXT 10.1007/978-3-319-07254-8_1

1. Random Numbers

Francesca Biagini 1

(1)

Department of Mathematics, Ludwig-Maximilians Universitt, Munich, Germany

(2)

Department of Mathematics, Universit di Bologna, Bologna, Italy

Francesca Biagini (Corresponding author)

Email:

Massimo Campanino

Email:

1.1 Introduction

Probability Theory deals with the quantification of our degree of uncertainty. Its main object of interest are random entities and, in particular, random numbers. What is meant by random number ?

A random number is a well defined number, whose value is not necessarily known. For example we can use random numbers to describe the result of a determined experiment, or the value of an option at a prefixed time, or the value of a meteorological magnitude at a given time. All these quantities have a well defined value, but may not be known either because they refer to the future and there are no means to predict their values with certainty or, even if they refer to the past, there is no available information at the moment.

We shall denote random numbers with capital letters. Even if the value of a random number is in general not known, we can speak about the set of its possible values , that will be denoted by I ( X ). Certain numbers can be considered as particular cases of random numbers, whose set of possible values consists of a single element.

Example 1.1.1

Let the random numbers X , Y represent respectively the results of throwing a coin and a die. If we denote head and tail by 0 and 1 and the sides of the die with the numbers from 1 to 6, we have:

The random number X is:

• upper bounded if I(X) is upper bounded ( );
• lower bounded if I(X) is lower bounded ( );
• bounded if I(X) is both upper and lower bounded ( , ).

Given two random numbers X and Y, we denote by I ( X , Y ) the set of pairs of values that ( X , Y ) can attain. In general given n random numbers , we denote by the set of possible values that can attain.

The random numbers X and Y are said to be logically independent if

where denotes the Cartesian product of I ( X ) and I ( Y ).

Similarly the random numbers are said to be logically independent if .

Example 1.1.2

In a lottery two balls are consecutively drawn without substitution from an urn that contains 90 balls numerated from 1 to 90. Let X and Y represent the random numbers corresponding respectively to the first and the second drawing. The set of possible pairs is then

Clearly as I ( X , Y ) does not contain pairs of the type ( i , i ), with . The random numbers X and Y therefore are not logically independent.

By using random numbers we can perform usual arithmetic operations, obtaining again random numbers. We introduce the following operations that we will apply to random numbers. For real x and y

;

;

.

As it is easy to verify, these operations satisfy the following properties:

distributive property

(1.1)

(1.2)

associative property

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