Quasi-continuous random variables and processes under the G-expectation framework

Formal Reasoning about Expectation Properties for Continuous Random Variables

Expectation (average) properties of continuous random variables are widely used to judge performance characteristics in engineering and physical sciences. This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random variables in a theorem prover. Starting from the relatively complex higher-order-logic definition of expec...

Random Variables , Distributions and Expectation 1 Random Variables

We’ve used probablity to model a variety of experiments, games, and tests. Throughout, we have tried to compute probabilities of events. We asked, for example, what is the probability of the event that you win the Monty Hall game? What is the probability of the event that it rains, given that the weatherman carried his umbrella today? What is the probability of the event that you have a rare di...

Continuous Random Variables

Math 394 1 (Almost bullet-proof) Definition of Expectation Assume we have a sample space Ω, with a σ−algebra of subsets F , and a probability P , satisfying our axioms. Define a random variable as a a function X : Ω → R, such that all subsets of Ω of the form {ω |a < X(ω) ≤ b}, for any real a ≤ b are events (belong to F). Assume at first that the range of X is bounded, say it is contained in th...

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...

Random Variables and Stochastic Processes