The Natural Axiom System of Probability Theory

The following sections are included:

• Preface

• Preface of Chinese Edition

• Contents

• Preliminary

The following sections are included:

• Research object

• Research task

• Probability
  • Classical definition of probability
  • Geometric probability
  • Frequency illustration of probability
  • Subjective illustration of probability
  • Prima facie of the application

The following sections are included:

• Element of probability theory and six groups of axioms

• The first group of axioms: the group of axioms of event space
  • Event space
  • Sure event and impossible event
  • Rules of operations
  • Event fields and event σ-fields
  • Remark

• The second group of axioms: the group of axioms of causal space
  • The questions and the methods
  • Intuitive background of the causal point
  • The group of axioms of causal space
  • Remark

• The third group of axioms: the group of axioms of random test
  • Intuitive background
  • The group of axioms of random test
  • Random sub-tests
  • Modelling of random test(1): Listing method
  • Modelling of random test(2): Expansionary method
  • Remark

• Several kinds of typical random tests
  • σ-isomorphism; Standard tests
  • n-type tests
  • Countable type tests
  • n-dimensional É. Borel tests
  • Countable dimensional and any dimensional Borel tests
  • Remark

• Joint random tests
  • Joint event σ-fields
  • Joint random tests
  • Product random tests
  • Duplicate random tests
  • n-dimensional joint tests
  • Countable dimensional and any dimensional joint tests
  • Remark

• The forth group of axioms: the group of axioms of probability measure
  • Intuitive background
  • The group of axioms of probability measure
  • Existence of probability measure
  • Probability spaces of classical type
  • Probability spaces of geometric type
  • Whether do the frequency illustration and subjection illustration satisfy the fourth group of axioms?
  • Basic properties of probability
  • Extension of probability measure
  • Independent product probability space
  • Remark

• Point functions on random test
  • Probability spaces of discrete type
  • Kolmogorov's probability spaces
  • Kolmogorov's probability spaces: n-dimensional Case
  • Kolmogorov's probability spaces: infinite dimensional case
  • Remark

• The fifth group of axioms: the group of axioms of conditional probability measure
  • Intuitive background
  • The group of axioms of conditional probability measure
  • Independence of two sub-tests
  • Independence of n sub-tests
  • Independence of a family of sub-tests
  • Basic theorems of conditional probability values
  • Remark

• Point functions on random test (continued)
  • Conditional density
  • Conditional density-probability
  • Point functions on the probability space of discrete type
  • Point functions on n-dimension Kolmogorov's probability space
  • Remark

• The sixth group of axioms: the group of axioms of probability modeling
  • The group of axioms of probability modeling
  • Modelling of probability space of classical type
  • Modelling of probability space of discrete type
  • Modelling of probability space of geometrical Type
  • Modelling of n-dimension Kolmogorov's probability space
  • Modelling of independent product probability Space
  • Importance of the axiom of frequency

The following sections are included:

• Intuitive background of random variables
  • Intuitive background: the situation of probability space to be known
  • Importance of the distribution law
  • Intuitive background: the situation of probability space to be not known
  • Importance of the distribution function
  • Universality of the random variables in application
  • Universality of the random variables in theory

• Basic conceptions of random variable
  • Representation probability space and the digital characters
  • Value probability space and the distribution function
  • Remark: probability analytic method

• Basic conceptions of random vector
  • Representation probability space and the digital characters
  • Value probability space and the distribution function
  • Random sub-vectors and the marginal distribution functions
  • Independence
  • Various conditional probabilities
  • Conditional mathematical expectation of X given sub-test
  • Conditional mathematical expectation of X given Y
  • Remark

• Basic conceptions of broad stochastic process
  • Representation probability space and the digital characters
  • Value probability space and the family of finite dimensional distribution functions
  • Family of finite dimension random vectors
  • Index set J

The following sections are included:

• Bibliography

• Index