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:
“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)
“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)
“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)