Unformatted text preview: omplementation and countable unions
and intersections. An understanding of how to construct this extension in general requires
measure theory but is not necessary for our purposes.
The simplest and most natural way of creating a probability measure for this extended sample space and class of events is through the assumption that the repetitions are statistically
independent of each other. More precisely, we assume that for each of the extended events
(A1 , A2 , . . . , An ) contained in ≠×n , we have
Pr {(A1 , A2 , . . . , An )} = Yn i=1 Pr {Ai } , (1.14) where Pr {Ai } is the probability of event Ai in the original model. Note that since ≠ can
be substituted for any Ai in this formula, the subset condition of (1.11) is automatically
satisﬁed. In other words, for any probability model, there is an extended independent nrepetition model for which the events in each repetition are independent of those in the other
repetitions. In what follows, we refer to this as the probability model for n independent
identically distributed (IID) experiments.
The niceties of how this model for n IID experiments is created depend on measure theory,
but we can rely on the fact that such a model can be created, and that the events in each
repetition are independent of those in the other repetitions. What we have done here is
very important conceptually. A probability model for an experiment does not say anything
directly about repeated experiments. However, questions about independent repeated experiments can be handled directly within this n IID experiment model. This can also be
extended to a countable number of IID experiments. 1.3. PROBABILITY REVIEW 1.3.3 11 Random variables The outcome of a probabilistic experiment often speciﬁes a collection of numerical values
such as temperatures, voltages, numbers of arrivals or departures in various intervals, etc.
Each such numerical value varies, depending on the particular outcome of the experiment,
and thus can be viewed as a mapping from the set ≠ of sample points to the set R of real
numbers. Note that R does not include ±1, and in those rare instances where ±1 should
be included, it is called the extended set of real numbers. These mappings from sample
points to numerical values are called random variables.
Deﬁnition 1.4. A random variable (rv) is a function X from the sample space ≠ of a
probability model to the set of real numbers R. That is, each sample point ω in ≠, except
perhaps for a subset11 of probability 0, is mapped into a real number denoted by X (ω ). There
is an additional technical requirement that, for each real x, the set {X (ω ) ≤ x} must be an
event in the probability model.
As with any function, there is often some confusion between the function itself, which is
called X in the deﬁnition above, and the value X (ω ) the function takes on for a sample
point ω . This is particularly prevalent with random variables (rv’s) since we intuitively
associate a rv with its sample value when an experiment is performed. We try to control
that confusion here by using X , X (ω ), and x respectively, to refer to the rv, the sample
value taken for a given sample point ω , and a generic sample value.
Deﬁnition 1.5. The distribution function12 FX (x) of a random variable (rv) X is a function, R → R, deﬁned by FX (x) = Pr {ω ∈ ≠  X (ω ) ≤ x}. The argument ω is usual ly
omitted for brevity, so FX (x) = Pr {X ≤ x}.
Note that x is the argument of FX (x), whereas the subscript X denotes the particular
rv under consideration. As illustrated in Figure 1.1, the distribution function FX (x) is
nondecreasing with x and must satisfy the limits limx→−1 FX (x) = 0 and limx→1 FX (x) =
1 (see Exercise 1.3).
The concept of a rv is often extended to complex random variables (rv’s) and vector rv’s.
A complex random variable is a mapping from the sample space to the set of ﬁnite complex
numbers, and a vector random variable (rv) is a mapping from the sample space to the
ﬁnite vectors in some ﬁnite dimensional vector space. Another extension is that of defective
rvs. X is defective if there is an event of positive probability for which the mapping is
either undeﬁned or deﬁned to be either +1 or −1. When we refer to random variables in
this text (without any modiﬁer such as complex, vector, or defective), we explicitly restrict
attention to the original deﬁnition, i.e., a function from ≠ to R.
If X has only a ﬁnite or countable number of possible sample values, say x1 , x2 , . . . , the
probability Pr {X = xi } of each sample value xi is called the probability mass function
11
We will ﬁnd many situations, particularly when taking limits, when rv’s are undeﬁned for some set
of sample points of probability zero. These are still called rv’s (rather than defective rv’s ) and the zero
probability subsets can usually be ignored.
12
Some people refer to the distribution function as the cumulative distribution function. 12 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY 1
FX (x) 0 Figure 1.1: Example of a distribution function for a rv that is neith...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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