Definition
The requirements for a function μ to be a probability measure on a probability space are that:
-
- μ must return results in the unit interval, returning 0 for the empty set and 1 for the entire space.
-
- μ must satisfy the countable additivity property that for all countable collections of pairwise disjoint sets:
For example, given three elements 1, 2 and 3 with probabilities 1/4, 1/4 and 1/2, the value assigned to {1, 3} is 1/4 + 1/2 = 3/4, as in the diagram on the right.
The conditional probability based on the intersection of events defined as:
satisfies the probability measure requirements so long as is not zero.
Probability measures are distinct from the more general notion of fuzzy measures in which there is no requirement that the fuzzy values sum up to 1, and the additive property is replaced by an order relation based on set inclusion.
Read more about this topic: Probability Measure
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“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)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)