Page 87 - Fundamentals of Communications Systems

Chapter
3

Review of Probability and

Random Variables

This chapter is intended to review introductory material on random variables,
typically covered in an undergraduate curriculum. The most important use of
this chapter will be the introduction of the notation used in this text. More proofs
will be provided in this chapter than the last chapter because typically this
subject matter is not as synthesized at the start of a senior level communications
course. Texts that give a more detailed treatment of the subject of probability
and random variables are [DR87, LG89, Hel91, CM86, Dev00, Sti99, YG98].

3.1 Axiomatic Definitions of Probability
Characterization of random events or experiments is critical for communication
system design and analysis. A majority of the analyses of random events or
experiments are extensions of the simple axioms or definitions presented in this
section. A random experiment is characterized by a probability space consisting
of a sample space , a field F, and a probability measure P (•). This probability
space will be denoted ( , F, P ).

EXAMPLE 3.1
Consider the random experiment of rolling a fair dice
={1, 2, 3, 4, 5, 6} (3.1)
The field, F, is the set of all possible combinations of outputs, i.e., consider the fol-
lowing outcomes A 1 ={the die shows a 1}, A 2 ={the die shows an even number}, A 3 =
{the die shows a number less than 4}, and A 4 ={the die shows an odd number}, which
implies
A 1 , A 2 , A 3 , A 4 ∈ F (3.2)

3.1
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