Conditional density
From Encyclopedia of Mathematics
The density of a conditional distribution. Let be a probability space, let
be the
-algebra of Borel sets on the line, let
be a sub-
-algebra of
, let
![]() |
be the conditional distribution of with respect to
, and let
![]() |
be the conditional distribution function of with respect to
. If
![]() |
then is called the conditional density of the distribution of
with respect to the
-algebra
.
If and
are random variables,
is the density of the distribution of
and
is the joint density of the distribution of
and
, then
![]() |
defines the conditional density of the distribution of the random variable for fixed values
of
for which
.
References
[1] | Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) |
How to Cite This Entry:
Conditional density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_density&oldid=15893
Conditional density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_density&oldid=15893
This article was adapted from an original article by V.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article