Conditional distribution
A function of an elementary event and a Borel set, which for each fixed elementary event is a probability distribution and for each fixed Borel set is a conditional probability.
Let be a probability space,
the
-algebra of Borel sets on the line,
a random variable defined on
and
a sub-
-algebra of
. A function
defined on
is called a (regular) conditional distribution of the random variable
with respect to the
-algebra
if:
a) for fixed the function
is
-measurable;
b) with probability one, for fixed the function
is a probability measure on
;
c) for arbitrary ,
![]() |
Similarly one can define the conditional distribution of a random element with values in an arbitrary measurable space
. If
is a complete separable metric space and
is the
-algebra of Borel sets, then the conditional distribution of the random element
relative to any
-algebra
,
, exists.
The function is called the conditional distribution function of the random variable
with respect to the
-algebra
.
The conditional distribution (conditional distribution function) of a random variable with respect to a random variable
is defined as the conditional distribution (conditional distribution function) of
with respect to the
-algebra generated by
.
The conditional distribution function of a random variable
with respect to
is a Borel function of
; for
its value
is called the conditional distribution function of
for a fixed value of
. If
has a density
, then
![]() |
where is the joint distribution function of
and
.
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] | M. Loève, "Probability theory" , Princeton Univ. Press (1963) |
[3] | I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1974) (Translated from Russian) |
Comments
Another definition of a conditional distribution is as a function of a regular event and a Borel set such that, for fixed
,
is a probability measure and, for fixed
,
is a measurable function.
References
[a1] | L.P. Breiman, "Probability" , Addison-Wesley (1968) |
Conditional distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_distribution&oldid=11344