Page 25 - Elements of Distribution Theory

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1.4 Distribution Functions 11

Proof. Consider the experiment with = (0, 1) and, suppose that, for any set A ∈ ,
P(A)given by

P(A) = dx.
A
Given a function F satisfying (DF1)–(DF3), define a random variable X by
X(ω) = inf{x ∈ R: F(x) ≥ ω}.

Then

Pr(X ≤ x) = P({ω ∈ : X(ω) ≤ x}) = P({ω ∈ : ω ≤ F(x)}) = F(x).
Hence, F is the distribution function of X.

The distribution function is a useful way to describe the probability distribution of a
random variable. The following theorem states that the distribution function of a random
variable X completely characterizes the probability distribution of X.

Theorem 1.4. If two random variables X 1 ,X 2 each have distribution function F, then X 1
and X 2 have the same probability distribution.

A detailed proof of this result is beyond the scope of this book; see, for example, Ash
(1972, Section 1.4) or Port (1994, Section 10.3).
It is not difficult, however, to give an informal explanation of why we expect such a
result to hold. The goal is to show that, if X 1 and X 2 have the same distribution function,
then, for ‘any’ set A ⊂ R,
Pr(X 1 ∈ A) = Pr(X 2 ∈ A).

First suppose that A is an interval of the form (a 0 , a 1 ]. Then
Pr(X j ∈ A) = F(a 1 ) − F(a 0 ), j = 1, 2

c
so that Pr(X 1 ∈ A) = Pr(X 2 ∈ A). The same is true for A .Now consider a second interval
B = (b 0 , b 1 ]. Then

∅ if b 0 > a 1 or a 0 > b 1

 B if a 0 ≤ b 0 < b 1 ≤ a 1


A ∩ B = A if b 0 ≤ a 0 < a 1 ≤ b 1 .
 (a 0 , b 1 ] if b 1 ≤ a 1 and b 0 ≤ a 0



(b 0 , a 1 ] if a 1 ≤ b 1 and a 0 ≤ b 0
In each case, A ∩ B is an interval and, hence, Pr(X j ∈ A ∩ B) and Pr(X j ∈ A ∪ B)donot
depend on j = 1, 2. The same approach can be used for any finite collection of intervals.
Hence, if a set is generated from a finite collection of intervals using set operations such as
union, intersection, and complementation, then Pr(X 1 ∈ A) = Pr(X 2 ∈ A).
However, we require that this equality holds for ‘any’ set A.Of course, we know that
probability distibutions cannot, in general, be defined for all subsets of R. Hence, to pro-
ceed, we must pay close attention to the class of sets A for which Pr(X 1 ∈ A)is defined.
Essentially, the result stated above for a finite collection of intervals must be extended to