regular conditional probability
Suppose is a probability space and be an event with . It is easy to see that defined by
is clearly non-negative;
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 ).
for each event , is a conditional probability (http://planetmath.org/ProbabilityConditioningOnASigmaAlgebra) (as a random variable) of given ; that is,
is -measurable (http://planetmath.org/MathcalFMeasurableFunction) and
for every , we have
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|
|Date of creation||2013-03-22 16:25:24|
|Last modified on||2013-03-22 16:25:24|
|Last modified by||CWoo (3771)|