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Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence , and it is connected with free products .
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.
There are 42 noncrossing and 10 crossing partitions of a 5-element set The 14 noncrossing partitions of a 4-element set ordered by refinement in a Hasse diagram. In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability.
Downward-lexicographic dominance, denoted , means that has a larger probability than of returning the best outcome, or both and have the same probability to return the best outcome but has a larger probability than of returning the second-best best outcome, etc. Upward-lexicographic dominance is defined analogously based on the probability to ...
In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a noncentral generalization of the (ordinary) F-distribution. It describes the distribution of the quotient ( X / n 1 )/( Y / n 2 ), where the numerator X has a noncentral chi-squared distribution with n 1 degrees of freedom and ...
The term non-parametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance. A histogram is a simple nonparametric estimate of a probability distribution. Kernel density estimation is another method to estimate a probability distribution.
The total variation distance (or half the norm) arises as the optimal transportation cost, when the cost function is (,) =, that is, ‖ ‖ = (,) = {(): =, =} = [], where the expectation is taken with respect to the probability measure on the space where (,) lives, and the infimum is taken over all such with marginals and , respectively.
This probability density is calculable from the Survival probability (a more common probability measure in statistics). Consider the absorbing boundary condition p ( x c , t ) = 0 {\displaystyle p(x_{c},t)=0} (The subscript c for the absorption point x c {\displaystyle x_{c}} is an abbreviation for cliff used in many texts as an analogy to an ...