Page 34 - Elements of Distribution Theory

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20 Properties of Probability Distributions

1.6 Density and Frequency Functions

Consider a real-valued random variable X with distribution function F and range X. Sup-
pose there exists a function p : R → R such that
x
F(x) = p(t) dt, −∞ < x < ∞. (1.1)
−∞
The function p is called the density function of the distribution or, more simply, of X; since
F is nondecreasing, p can be assumed to be nonnegative and we must have
∞

p(x) dx = 1.
−∞
We also assume that any density function is continuous almost everywhere and is of
bounded variation, which ensures that the Riemann integral of the density function exists;
see Sections A3.1.8, A.3.3.3, and A3.4.9 of Appendix 3.
In this case, it is clear that the distribution function F must be a continuous function; in
fact, F is an absolutely continuous function. Absolute continuity is stronger than ordinary
continuity; see Appendix 1 for further discussion of absolutely continuous functions. Hence,
when (1.1) holds, we say that the distribution of X is an absolutely continuous distribution;
alternatively, we say that X is an absolutely continuous random variable.
Conversely, if F is an absolutely continuous function, then there exists a density func-
tion p such that (1.1) holds. In many cases, the function p can be obtained from F by the
fundamental theorem of calculus; see Theorem 1.9 below. It is important to note, however,
that the density function of a distribution is not uniquely defined. If

p 1 (x) = p 2 (x) for almost all x,
and
x
F(x) = p 1 (t) dt, −∞ < x < ∞,
−∞
then
x
F(x) = p 2 (t) dt, −∞ < x < ∞;
−∞
see Section A3.1.8 of Appendix 3 for discussion of the term “almost all.” In this case, either
p 1 or p 2 may be taken as the density function of the distribution. Generally, we use the
version of the density that is continuous, if one exists.
The following theorem gives further details on the relationship between density and
distribution functions.

Theorem 1.9. Let F denote the distribution function of a distribution on R.
(i) Suppose that F is absolutely continuous with density function p. If p is continuous
at x then F (x) exists and p(x) = F (x).

(ii) Suppose F (x) exists for all x ∈ R and

∞

F (x) dx < ∞.
−∞

Then F is absolutely continuous with density function F .