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3 1

Random variable

Expectation Independence

General definitions

Let the probability space Q,;j/t,

gJ

be given. @il 00, + 0) the finite)

real line, 2l?

=

[-00,

+00]

the extended real line, qjI

=

the Euclidean Borel

field on

91

1

,

JI3

= the extended Borel field. A set in 9 3*

IS

just a set in

0: 3

possibly enlarged by one or both points oo

DEFINITION OF A RANDOM VARIABLE. A real, extended-valued random varI

able is a function X whose domain

is

a set in

dr

and whose range is

contained in ?J?

=

[-00,

+00]

such that for each

B

in a 3*, we have

1)

{w:X w) E

B}

E n ~ r

where n 1fr is the trace of

;y,

on

~

A complex-valued random variable is

a function on a set in

;: r

to the complex plane whose real and imaginary

parts are both real, finite-valued random variables.

This definition in its generality is necessary for logical reasons in many

applications, but for a discussion of basic properties we may suppose =

Q

and that

X

is real andfinite valued with probability one.

This restricted meaning

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GENERAL DEFINITIONS I

35

of

a random variable , abbreviated as r.v. , will be understood in the book

unless otherwise specified. The general case may be reduced to this one by

considering the trace of Q,;jh ,q: on

or on

the domain of finiteness

~ o = {w: IX w)1

0 X-

I .

This p is called the probability distribution measure or p.m. of X, and its

associated d.f.

F

according to Theorem 2.2.4 will be called the d.f. of X.