Page 41 - Elements of Distribution Theory

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1.7 Integration with Respect to a Distribution Function 27

more generally, for any set A ⊂ R,we should have

Pr(X ∈ A) = dF(x).
A
It is also natural to expect that any definition of an integral satisfies certain linearity and
nonnegativity conditions:

g 2 (x) dF(x)
[a 1 g 1 (x) + a 2 g 2 (x)] dF(x) = a 1 g 1 (x) dF(x) + a 2
R R R
and

g(x) dF(x) ≥ 0 whenever g(x) ≥ 0 for all x.
R
These properties, together with a continuity property, can be used to give a precise definition
to integrals of the form

g(x) dF(x);
R
see Appendix 1 for further details.
Although this type of integral can be computed for any distribution function F, the
computation is particularly simple if the distribution is either absolutely continuous or
discrete. Suppose X has an absolutely continuous distribution with density p. Then

x x
F(x) = dF(t) = p(t) dt;
−∞ −∞
this result generalizes to

∞ ∞

g(x) dF(x) = g(x)p(x) dx,
−∞ −∞
provided that the integrals exist. If X has discrete distribution with frequency function p,a
similar result holds, so that
∞

g(x) dF(x) = g(x)p(x),
−∞
x∈X
provided that the sum exists.
Thus, if attention is restricted to random variables with either an absolutely continuous
or discrete distribution, then
∞

∞ 
g(x)p(x)dx if X is absolutely continuous
−∞
g(x) dF(x) = ; (1.2)
−∞  g(x)p(x) if X is discrete
x∈X
here p represents either the density function or the frequency function of the distribution.
The case in which the distribution is neither absolutely continuous nor discrete may be
d
viewed as an extension of these results. If X is a random vector taking values in R , then
(1.2) generalizes in the obvious manner:

d g(x)p(x) dx if X is absolutely continuous
∞  R

g(x) dF(x) = .
−∞  g(x)p(x) if X is discrete
x∈X