Page 9 - Probability, Random Variables and Random Processes

Chapter 1

Probability

1.1 INTRODUCTION
The study of probability stems from the analysis of certain games of chance, and it has found
applications in most branches of science and engineering. In this chapter the basic concepts of prob-
ability theory are presented.

1.2 SAMPLE SPACE AND EVENTS
A. Random Experiments:

In the study of probability, any process of observation is referred to as an experiment. The results
of an observation are called the outcomes of the experiment. An experiment is called a random experi-
ment if its outcome cannot be predicted. Typical examples of a random experiment are the roll of a
die, the toss of a coin, drawing a card from a deck, or selecting a message signal for transmission from
several messages.

B. Sample Space:
The set of all possible outcomes of a random experiment is called the sample space (or universal
set), and it is denoted by S. An element in S is called a sample point. Each outcome of a random
experiment corresponds to a sample point.

EXAMPLE 1.1 Find the sample space for the experiment of tossing a coin (a) once and (b) twice.
(a) There are two possible outcomes, heads or tails. Thus
S = {H, T)
where H and T represent head and tail, respectively.
(b) There are four possible outcomes. They are pairs of heads and tails. Thus
S = (HH, HT, TH, TT)

EXAMPLE 1.2 Find the sample space for the experiment of tossing a coin repeatedly and of counting the number
of tosses required until the first head appears.
Clearly all possible outcomes for this experiment are the terms of the sequence 1,2,3, . . . . Thus
s = (1, 2, 3, . . .)
Note that there are an infinite number of outcomes.

EXAMPLE 1.3 Find the sample space for the experiment of measuring (in hours) the lifetime of a transistor.
Clearly all possible outcomes are all nonnegative real numbers. That is,
S=(z:O<z<oo}
where z represents the life of a transistor in hours.
Note that any particular experiment can often have many different sample spaces depending on the observ-
ation of interest (Probs. 1.1 and 1.2). A sample space S is said to be discrete if it consists of a finite number of