Search results
Results from the WOW.Com Content Network
Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity. For a given fuzzy set A {\displaystyle A} , its complement ¬ A {\displaystyle \neg {A}} (sometimes denoted as A c {\displaystyle A^{c}} or c A {\displaystyle cA} ) is defined by the following ...
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
The two equivalent definitions are as follows. Using union and intersection: define [1] [2] = and = If these two sets are equal, then the set-theoretic limit of the sequence exists and is equal to that common set. Either set as described above can be used to get the limit, and there may be other means to get the limit as well.
Another example of events being collectively exhaustive and mutually exclusive at same time are, event "even" (2,4 or 6) and event "odd" (1,3 or 5) in a random experiment of rolling a six-sided die. These both events are mutually exclusive because even and odd outcome can never occur at same time.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. This inequality provides an upper bound on the probability of occurrence of at least one ...
An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers.It is the algebra on which the Borel measure is defined. . Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel a
The situation that appears in the derangement example above occurs often enough to merit special attention. [7] Namely, when the size of the intersection sets appearing in the formulas for the principle of inclusion–exclusion depend only on the number of sets in the intersections and not on which sets appear. More formally, if the intersection