Definition of Conditional Probability
For any event, define the indicator function:
which is a random variable with respect to the Borel σ-algebra on (0,1). Note that the expectation of this random variable is equal to the probability of A itself:
Then the conditional probability given is a function such that is the conditional expectation of the indicator function for A:
In other words, is a -measurable function satisfying
A conditional probability is regular if is also a probability measure for all ω ∈ Ω. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.
- For the trivial sigma algebra the conditional probability is a constant function,
- For, as outlined above, .
Read more about this topic: Conditional Expectation
Famous quotes containing the words definition of, definition, conditional and/or probability:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“The population of the world is a conditional population; these are not the best, but the best that could live in the existing state of soils, gases, animals, and morals: the best that could yet live; there shall be a better, please God.”
—Ralph Waldo Emerson (1803–1882)
“Legends of prediction are common throughout the whole Household of Man. Gods speak, spirits speak, computers speak. Oracular ambiguity or statistical probability provides loopholes, and discrepancies are expunged by Faith.”
—Ursula K. Le Guin (b. 1929)