Page 40 - Elements of Distribution Theory

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26 Properties of Probability Distributions

Random vectors
The concepts of discrete and absolutely continuous distributions can be applied to a vector-
valued random variable X = (X 1 ,..., X d )as well. If the distribution function F of X may
be written
x d x 1

F(x 1 ,..., x d ) = ··· p(t) dt
−∞ −∞
d
for some function p on R , then X is said to have an absolutely continuous distribution
with density p.If the range of X is a countable set X with
p(x) = Pr(X = x), x ∈ X,
then X is said to have a discrete distribution with frequency function p.

Example 1.25 (Uniform distribution on the unit cube). Let X denote a three-dimensional
3
random vector with the uniform distribution on (0, 1) ,as defined in Example 1.7. Then,
for any A ∈ R 3

Pr(X ∈ A) = dt 1 dt 2 dt 3 .
A∩(0,1) 3
Hence, X has an absolutely continuous distribution with density function
3
p(x) = 1, x ∈ (0, 1) .

Example 1.26 (A discrete random vector). Let X = (X 1 , X 2 ) denote a two-dimensional
random vector such that
1 1 1
Pr(X = (0, 0)) = , Pr(X = (1, 0)) = , and Pr(X = (0, 1)) = .
2 4 4
Then X isadiscreterandomvariablewithrange{(0, 0), (0, 1), (1, 0)}andfrequencyfunction
1 (x 1 + x 2 )
p(x 1 , x 2 ) = − , x 1 = 0, 1; x 2 = 0, 1.
2 4

1.7 Integration with Respect to a Distribution Function
Integrals with respect to distribution functions, that is, integrals of the form

g(x) dF(x),
R d
play a central role in distribution theory. For readers familiar with the general theory of
integration with respect to a measure, the definition and properties of such an integral
d
follow from noting that F defines a measure on R .In this section, a brief description of
such integrals is given for the case in which X is a real-valued random variable; further
details and references are given in Appendix 1.
Suppose X is a real-valued random variable with distribution function F. Then we expect
that
x
F(x) = dF(t);
−∞