Conditional probability

The conditional probability of an event relative to another event is a characteristic connecting the two events. If and are events and , then the conditional probability of the event relative to (or under the condition, or with respect to) is defined by the equation

The conditional probability can be regarded as the probability that the event is realized under the condition that has taken place. For independent events and the conditional probability coincides with the unconditional probability .

About the connection between the conditional and unconditional probabilities of events see Bayes formula and Complete probability formula.

The conditional probability of an event with respect to a -algebra is a random variable , measurable relative to , for which

for any . The conditional probability with respect to a -algebra is defined up to equivalence.

If the -algebra is generated by a countable number of disjoint events having positive probability and the union of which coincides with the whole space , then

The conditional probability of an event with respect to the -algebra can be defined as the conditional mathematical expectation of the indicator function of .

Let be a probability space and let be a subalgebra of . The conditional probability is called regular if there exists a function , , , such that

a) for a fixed the function is a probability on the -algebra ;

b) with probability one.

For a regular conditional probability the conditional mathematical expectation can be expressed by integrals, with the conditional probability taking the role of the measure.

The conditional probability with respect to a random variable is defined as the conditional probability with respect to the -algebra generated by .

References

[1]	A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)
[2]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[3]	M. Loève, "Probability theory" , Princeton Univ. Press (1963)