Page 28 - Elements of Distribution Theory

P1: JZP
052184472Xc01 CUNY148/Severini May 24, 2005 17:52

14 Properties of Probability Distributions

d
(ii) If x n = (x n1 ,..., x nd ),n = 1, 2,... is a sequence in R such that each sequence
x nj ,n = 1, 2,... is a decreasing sequence with limit x j ,j = 1,..., d, then
lim F(x n ) = F(x).
n→∞
d
Proof. Let x = (x 1 , x 2 ,..., x d ) and y = (y 1 , y 2 ,..., y d ) denote elements of R satisfying
the condition in part (i) of the theorem. Define
A = (−∞, x 1 ] × ··· × (−∞, x d ]

and

B = (−∞, y 1 ] × ··· × (−∞, y d ].
Then F(x) = P X (A), F(y) = P X (B) and part (i) of the theorem follows from the fact that
A ⊂ B.
For part (ii), define

A n = (−∞, x n1 ] ×· · · (−∞, x nd ], n = 1, 2,....
Then A 1 ⊃ A 2 ⊃· · · and

∞
∩ n=1 A n = (−∞, x 1 ] × ··· × (−∞, x n ].
The result now follows from (P5).

We saw that the probability that a real-valued random variable takes values in a set (a, b]
can be expressed in terms of its distribution, specifically,

Pr(a < X ≤ b) = F(b) − F(a).
A similar result is available for random vectors, although the complexity of the expression
increases with the dimension of the random variable. The following example illustrates the
case of a two-dimensional random vector; the general case is considered in Theorem 1.7
below.

Example 1.14 (Two-dimensional random vector). Let X = (X 1 , X 2 ) denote a two-
dimensional random vector with distribution function F. Consider the probability
Pr(a 1 < X 1 ≤ b 1 , a 2 < X 2 ≤ b 2 );

our goal is to express this probability in terms of F.
Note that
Pr(a 1 < X 1 ≤ b 1 , a 2 < X 2 ≤ b 2 ) = Pr(X 1 ≤ b 1 , a 2 < X 2 ≤ b 2 )
− Pr(X 1 ≤ a 1 , a 2 < X 2 ≤ b 2 )
= Pr(X 1 ≤ b 1 , X 2 ≤ b 2 ) − Pr(X 1 ≤ b 1 , X 2 ≤ a 2 )
− Pr(X 1 ≤ a 1 , X 2 ≤ b 2 ) + Pr(X 1 ≤ a 1 , X 2 ≤ a 1 )
= F(b 1 , b 2 ) − F(b 1 , a 2 ) − F(a 1 , b 2 ) + F(a 1 , a 2 ),
which yields the desired result.