probability conditioning on a sigma algebra
Let be a probability space and an event. Let be a sub sigma algebra of . The of given is defined to be the conditional expectation of the random variable defined on , given . We denote this conditional probability by . is also known as the indicator function.
Similarly, we can define a conditional probability given a random variable. Let be a random variable defined on . The conditional probability of given is defined to be , where is the sub sigma algebra of , generated by (http://planetmath.org/MathcalFMeasurableFunction) . The conditional probability of given is simply written .
Remark. Both and are random variables, the former is -measurable, and the latter is -measurable.
|Title||probability conditioning on a sigma algebra|
|Date of creation||2013-03-22 16:25:05|
|Last modified on||2013-03-22 16:25:05|
|Last modified by||CWoo (3771)|