Fuzzy Random Variables, Expectation, and Martingales

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...

Conditional Expectation and Martingales

These notes are intended to introduce concentration inequalities for martingales with bounded increments. The final section also provides a gentle introduction to conditional expectation based on sigma fields. In contrast to providing a firm foundation for measure-theoretic probability, the primary goal is to introduce the language and intuition used in the study of martingales. The presentatio...

Conditional Expectation and Martingales

This is a set of very hurriedly compiled notes. It has not been proof-checked and is likely to have some errors. These, though, should be minor errors and maybe cleared up by referring to the relevant texts. The texts that have been used in the preparation of the notes are Feller; Grimmett and Stirzaker; Goswami and Rao; David Williams; and Mitzenmacher and Upfal. The purpose behind the notes i...

Convergence Theorems for Generalized Random Variables and Martingales

We examine generalizations of random variables and martingales. We prove a new convergence theorem for setvalued martingales. We also generalize a well known characterization of set-valued random variables to the fuzzy setting. Keywords— Banach space, fuzzy, martingale, random variable, set-valued. 1 Set-valued random variables and martingales The theory of conditional expectations and martinga...

Dominated Convergence for Fuzzy Random Variables