8 1. Preparatory material
unaffected by modification on events of probability zero. Thus, a lack of
definedness on an event of probability zero will usually not cause difficulty,
so long as there are at most countably many such events in which one of
the probabilistic objects being studied is undefined. In such cases, one can
usually resolve such issues by setting a random variable to some arbitrary
value (e.g., 0) whenever it would otherwise be undefined.
We observe a few key subclasses and examples of random variables:
(i) Discrete random variables, in which R =
2R
is the discrete σ-
algebra, and R is at most countable. Typical examples of R include
a countable subset of the reals or complexes, such as the natural
numbers or integers. If R = {0, 1}, we say that the random variable
is Boolean, while if R is just a singleton set {c} we say that the
random variable is deterministic, and (by abuse of notation) we
identify this random variable with c itself. Note that a Boolean
random variable is nothing more than an indicator function I(E)
of an event E, where E is the event that the Boolean function
equals 1.
(ii) Real-valued random variables, in which R is the real line R and
R is the Borel σ-algebra, generated by the open sets of R. Thus
for any real-valued random variable X and any interval I, we have
the events “X I”. In particular, we have the upper tail event
“X λ” and lower tail event “X λ” for any threshold λ. (We
also consider the events “X λ” and “X λ” to be tail events;
in practice, there is very little distinction between the two types of
tail events.)
(iii) Complex random variables, whose range is the complex plane C
with the Borel σ-algebra. A typical event associated to a complex
random variable X is the small ball event “|X z| r” for some
complex number z and some (small) radius r 0. We refer to
real and complex random variables collectively as scalar random
variables.
(iv) Given an R-valued random variable X, and a measurable map f :
R R , the R -valued random variable f(X) is indeed a random
variable, and the operation of converting X to f(X) is preserved
under extension of the sample space and is thus probabilistic. This
variable f(X) can also be defined without reference to the sample
space as the unique random variable for which the identity
“f(X) S” = “X f
−1(S)”
holds for all R -measurable sets S.
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