regular conditional probability
Introduction
Suppose is a probability space and be an event with . It is easy to see that defined by
the conditional probability of event given , is a probability measure defined on , since:
-
1.
is clearly non-negative;
-
2.
;
-
3.
is countably additive: for if is a countable collection of pairwise disjoint events in , then
as is a collection of pairwise disjoint events also.
Regular Conditional Probability
Can we extend the definition above to , where is a sub sigma algebra of instead of an event? First, we need to be careful what we mean by , since, given any event , is not a real number. And strictly speaking, it is not even a random variable, but an equivalence class of random variables (each pair differing by a null event in ).
With this in mind, we start with a probability measure defined on and a sub sigma algebra of . A function is a called a regular conditional probability if it has the following properties:
-
1.
for each event , is a conditional probability (http://planetmath.org/ProbabilityConditioningOnASigmaAlgebra) (as a random variable) of given ; that is,
-
(a)
is -measurable (http://planetmath.org/MathcalFMeasurableFunction) and
-
(b)
for every , we have
-
(a)
-
2.
for every outcome , is a probability measure.
There are probability spaces where no regular conditional probabilities can be defined. However, when a regular conditional probability function does exist on a space , then by condition 2 of the definition, we can define a “conditional” probability measure on for each outcome in the sense of the first two paragraphs.
Title | regular conditional probability |
---|---|
Canonical name | RegularConditionalProbability |
Date of creation | 2013-03-22 16:25:24 |
Last modified on | 2013-03-22 16:25:24 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 60A99 |