Stochastic Relations of Random Variables and Processes

Stochastic relations of random variables and processes

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The m...

Random Variables and Stochastic Processes

Probability, random variables and stochastic processes

Generating Random Variables and Stochastic Processes

Some History • Earliest methods were manual – throwing dice, dealing cards, drawing balls from an urn • Mechanized devices in early 1900’s • Later, methods based on electric circuits • Many other schemes based on phone books, π etc. • Advent of computing led to interest in numerical and arithmetic methods – such methods generate numbers sequentially and are easily programmed – the most commonly...

Generating Random Variables and Processes

Many quantitative problems in science, engineering, and economics are nowadays solved via statistical sampling on a computer. Such Monte Carlo methods can be used in three different ways: (1) to generate random objects and processes in order to observe their behavior, (2) to estimate numerical quantities by repeated sampling, and (3) to solve complicated optimization problems through randomized...