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