Search results
Results from the WOW.Com Content Network
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
The following rules describe situations when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist. A ⪯ B {\displaystyle A\preceq B} if and only if for all non-decreasing functions u {\displaystyle u} , E [ u ( A ) ] ≤ E [ u ( B ) ] {\displaystyle \operatorname {E} [u(A ...
for all continuous bounded functions h. [2] Here E* denotes the outer expectation , that is the expectation of a “smallest measurable function g that dominates h ( X n ) ”. Properties
The set is called the index set [4] [51] or parameter set [28] [136] of the stochastic process. Often this set is some subset of the real line , such as the natural numbers or an interval, giving the set T {\displaystyle T} the interpretation of time. [ 1 ]
The word "bounded" makes no sense in a general topological space without a corresponding metric. Boundary is a distinct concept; for example, a circle (not to be confused with a disk) in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. A bounded set is not necessarily a closed set and vice
The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. [1]
Stochastic dominance is a partial order between random variables. [1] [2] It is a form of stochastic ordering.The concept arises in decision theory and decision analysis in situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble for a broad class of decision-makers.
are stochastically independent. [2] ... Then is called a random measure with independent S-increments, if for all bounded sets ,, …, and all the ...