probability density
The derivative of the distribution function corresponding to an absolutely-continuous probability measure.
Let
be a random vector taking values in an
-dimensional Euclidean space
, let
be its distribution function, and let there exist a non-negative function
such that
for any real
. Then
is called the probability density of
, and for any Borel set
,
Any non-negative integrable function
satisfy the condition
is the probability density of some random vector.
If two random vectors
and
taking values in
are independent and have probability densities
and
respectively, then the random vector
has the probability density
that is the convolution of
and
:
Let
and
be random vectors taking values in
and
and having probability densities
and
respectively, and let
be a random vector in
. If then
and
are independent,
has the probability density
, which is called the joint probability density of the random vectors
and
, where
 | (1) |
Conversely, if
has a probability density that satisfies (1), then
and
are independent.
The characteristic function
of a random vector
having a probability density
is expressed by
where if
is absolutely integrable then
is a bounded continuous function, and
The probability density
and the corresponding characteristic function
are related also by the following relation (Plancherel's identity): The function
is integrable if and only if the function
is integrable, and in that case
Let
be a measurable space, and let
and
be
-finite measures on
with
absolutely continuous with respect to
, i.e.
implies
,
. In that case there exists on
a non-negative measurable function
such that
for any
. The function
is called the Radon–Nikodým derivative of
with respect to
, while if
is a probability measure, it is also the probability density of
relative to
.
A concept closely related to the probability density is that of a dominated family of distributions. A family of probability distributions
on a measurable space
is called dominated if there exists a
-finite measure
on
such that each probability measure from
has a probability density relative to
(or, what is the same, if each measure from
is absolutely continuous with respect to
). The assumption of dominance is important in certain theorems in mathematical statistics.
References
[1] | Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) |
[2] | W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971) |
[3] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
How to Cite This Entry:
Density of a probability distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Density_of_a_probability_distribution&oldid=15505
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article