1 Last two lectures ¯ probability spaces ¯ probability measure ¯ random variables and stochastic processes ¯ distribution functions ¯ independence ¯ conditional probability ¯ memoriless property of geometric and exponential distributions ¯ expectation ¯ conditional expectation (double expectation) ¯ mean-square estimation 1

In this paper we investigate the notion of conditional independence and prove several information inequalities for conditionally independent random variables. Keywords— Conditionally independent random variables, common information, rate region.

Fuzziness is discussed in the context of multivalued lo&, and a corresponding view of fuzzy sets is given. Fuzzy random variables are introduced as random variables whose values are not real but fuzzy numbers, and subsequently redefined as a particular kind of fuzzy set. Expectations of fuzzy random variables, characteristic f~cti~ of fuzzy events, probabilities connected to fuzzy random variab...

The purpose of probability theory is to model random experiments so that we can draw inferences about them. The fundamental mathematical object is a triple (Ω,F , P ) called the probability space. A probability space is needed for each experiment or collection of experiments that we wish to describe mathematically. The ingredients of a probability space are a sample space Ω, a collection F of e...

In practice we are often faced with random experiments whose outcomes are not numbers (or vectors in &I”) but are expressed in inexact linguistic terms. As an example, consider a group of individuals chosen at random who are questioned about the weather on a particular city on a particular winter day. Some possible answers would be “cold,” “more or less cold,” “very cold,” “ extremely cold,” an...

In this paper, we discuss efficient exact random variate generation for the Bessel distribution. The expected time of the algorithm is uniformly bounded over all choices of the parameters, and the algorithm avoids any computation of Bessel functions or Bessel ratios.