Page 19 - Elements of Distribution Theory

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1.3 Random Variables 5

Define
k
˜
A k = A j , k = 1, 2,... ;
n=1
˜
˜
note that A 1 , A 2 ,... is an increasing sequence and that
˜
A 0 = lim A k .
k→∞
Hence, by (P4),
k
∞

˜
P(A 0 ) = lim P(A k ) = lim P(A n ) = P(A n ),
k→∞ k→∞
n=1 n=1
proving (P3). It follows that (P3) and (P4) are equivalent, proving the theorem.
1.3 Random Variables
Let ω denote the outcome of an experiment; that is, let ω denote an element of .In many
applications we are concerned primarily with certain numerical characteristics of ω, rather
d
than with ω itself. Let X : → X, where X is a subset of R for some d = 1, 2,..., denote
a random variable; the set X is called the range of X or, sometimes, the sample space of
X.For a given outcome ω ∈ , the corresponding value of X is x = X(ω). Probabilities
regarding X may be obtained from the probability function P for the original experiment.
Let P X denote a function such that for any set A ⊂ X,P X (A) denotes the probability that
X ∈ A. Then P X is a probability function defined on subsets of X and

P X (A) = P({ω ∈ : X(ω) ∈ A}).
WewillgenerallyusealessformalnotationinwhichPr(X ∈ A)denotesP X (A).Forinstance,
the probability that X ≤ 1 may be written as either Pr(X ≤ 1) or P X {(−∞, 1]}.In this book,
we will generally focus on probabilities associated with random variables, without explicit
reference to the underlying experiments and associated probability functions.
Note that since P X defines a probability function on the subsets of X,it must satisfy
conditions (P1)–(P3). Also, the issues regarding measurability discussed in the previous
section apply here as well.
d
When the range X of a random variable X is a subset of R for some d = 1, 2,..., it is
d
often convenient to proceed as if probability function P X is defined on the entire space R .
c
d
Then the probability of any subset of X is 0 and, for any set A ⊂ R ,
P X (A) ≡ Pr(X ∈ A) = Pr(X ∈ A ∩ X).
It is worth noting that some authors distinguish between random variables and random
d
vectors, the latter term referring to random variables X for which X is a subset of R for
d > 1. Here we will not make this distinction. The term random variable will refer to either
a scalar or vector; in those cases in which it is important to distinguish between real-valued
and vector random variables, the terms real-valued random variable and scalar random
variable will be used to denote a random variable with X ⊂ R and the term vector random
d
variable and random vector will be used to denote a random variable with X ⊂ R , d > 1.
Random vectors will always be taken to be column vectors so that a d-dimensional random