Definition
Let be a probability space, and let be a random variable, defined as a Borel-measurable function from to its state space Then a regular conditional probability is defined as a function called a "transition probability", where is a valid probability measure (in its second argument) on for all and a measurable function in E (in its first argument) for all such that for all and all
To express this in our more familiar notation:
where i.e. the topological support of the pushforward measure As can be seen from the integral above, the value of for points x outside the support of the random variable is meaningless; its significance as a conditional probability is strictly limited to the support of T.
The measurable space is said to have the regular conditional probability property if for all probability measures on all random variables on admit a regular conditional probability. A Radon space, in particular, has this property.
Read more about this topic: Regular Conditional Probability
Famous quotes containing the word definition:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)