Random Variables and Probability Distributions

2014 ◽  
pp. 63-106
Keyword(s):  
Probability Distributions ◽  
Random Variables

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

Introduction to Information Theory

2014 ◽  
Author(s):  
M. Vidyasagar
Keyword(s):  
Information Theory ◽  
Probability Distribution ◽  
Relative Entropy ◽  
Probability Distributions ◽  
Random Variables ◽  
Conditional Entropy ◽  
Random Variable ◽  
Entropy Function ◽  
Concave Functions ◽  
Leibler Divergence

This chapter provides an introduction to some elementary aspects of information theory, including entropy in its various forms. Entropy refers to the level of uncertainty associated with a random variable (or more precisely, the probability distribution of the random variable). When there are two or more random variables, it is worthwhile to study the conditional entropy of one random variable with respect to another. The last concept is relative entropy, also known as the Kullback–Leibler divergence, which measures the “disparity” between two probability distributions. The chapter first considers convex and concave functions before discussing the properties of the entropy function, conditional entropy, uniqueness of the entropy function, and the Kullback–Leibler divergence.

Continuous Random Numbers

2019 ◽  
pp. 54-69
Author(s):  
Robert H. Swendsen
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Delta Function ◽  
Ideal Gas ◽  
Dirac Delta Function ◽  
Bayesian Probability ◽  
Random Numbers ◽  
Continuous Variables ◽  
Dirac Delta ◽  
Discrete Random Variables

The theory of probability developed in Chapter 3 for discrete random variables is extended to probability distributions, in order to treat the continuous momentum variables. The Dirac delta function is introduced as a convenient tool to transform continuous random variables, in analogy with the use of the Kronecker delta for discrete random variables. The properties of the Dirac delta function that are needed in statistical mechanics are presented and explained. The addition of two continuous random numbers is given as a simple example. An application of Bayesian probability is given to illustrate its significance. However, the components of the momenta of the particles in an ideal gas are continuous variables.

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