Page 24 - Elements of Distribution Theory

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052184472Xc01 CUNY148/Severini May 24, 2005 17:52

10 Properties of Probability Distributions

(DF4) lim h→0 F(x − h) ≡ F(x−) = F(x) − Pr(X = x) = Pr(X < x).
+

Proof. Let a n , n = 1, 2,... denote any increasing sequence diverging to ∞ and let A n
∞
denote the event that X ≤ a n . Then P X (A n ) = F(a n ) and A 1 ⊂ A 2 ⊂· · · with ∪ A n equal
n=1
to the event that X < ∞.It follows from (P4) that
lim F(a n ) = Pr(X < ∞) = 1,
n→∞
establishing the first part of (DF1); the second part follows in a similar manner.
To show (DF2), let A 1 denote the event that X ≤ x 1 and A 2 denote the event that x 1 <
X ≤ x 2 . Then A 1 and A 2 are disjoint with F(x 1 ) = P X (A 1 ) and F(x 2 ) = P X (A 1 ∪ A 2 ) =
P X (A 1 ) + P X (A 2 ), which establishes (DF2).
For (DF3) and (DF4), let a n , n = 1, 2,..., denote any decreasing sequence converging
to 0, let A n denote the event that X ≤ x + a n , let B n denote the event that X ≤ x − a n , and
∞
let C n denote the event that x − a n < X ≤ x. Then A 1 ⊃ A 2 ⊃· · · and ∩ n=1 A n is the event
that X ≤ x. Hence, by (P5),
Pr(X ≤ x) ≡ F(x) = lim F(x + a n ),
n→∞
which establishes (DF3).
∞
Finally, note that F(x) = P X (B n ) + P X (C n ) and that C 1 ⊃ C 2 ⊃· · · with ∩ n=1 C n equal
to the event that X = x. Hence,
F(x) = lim F(x − a n ) + lim P X (C n ) = F(x−) + Pr(X = x),
n→∞ n→∞
yielding (DF4).

Thus, according to (DF2), a distribution function is nondecreasing and according to
(DF3), a distribution is right-continuous.
A distribution function F gives the probability of sets of the form (−∞, x]. The following
result gives expressions for the probability of other types of intervals in terms of F; the
proof is left as an exercise. As in Theorem 1.2, here we use the notation

F(x−) = lim F(x − h).
h→0 +

Corollary 1.1. Let X denote a real-valued random variable with distribution function F.
Then, for x 1 < x 2 ,
(i) Pr(x 1 < X ≤ x 2 ) = F(x 2 ) − F(x 1 )
(ii) Pr(x 1 ≤ X ≤ x 2 ) = F(x 2 ) − F(x 1 −)
(iii) Pr(x 1 ≤ X < x 2 ) = F(x 2 −) − F(x 1 −)
(iv) Pr(x 1 < X < x 2 ) = F(x 2 −) − F(x 1 )

Any distribution function possesses properties (DF1)–(DF4). Furthermore, properties
(DF1)–(DF3) characterize a distribution function in the sense that a function having those
properties must be a distribution function of some random variable.

Theorem 1.3. If a function F : R → [0, 1] has properties (DF1)–(DF3), then F is the
distribution function of some random variable.