Axiomatic Probability

Axiomatic Probability

  • Probability space
  • Conditional probability
  • Total probability equation, Bayes formular
  • Independence

Random Variable

Random Variable

  • Measurable function on probability space
  • Distribution: CDF, PDF, PMF
  • Integration (Expectation) - Riemann-Stieltjes integral
  • Characteristic function and moments

Function Analytic Approach to Probability

Conditioning and Dependence

  • Conditional expectation
  • Hierarchical models
  • Independence as an assumption and simplification.
  • Covariance and correlation

Special topics

Random Process

Note: Mathematical theories go complicated very fast; in fact, the description of random processes is already impractical and requires too much information. To put the mathematical model into practical use, vast simplification is needed.

Stochastic Analysis

Random Processes

(Ref: Scholtz, P266.)(Ref: Scholtz, P266.)

Nassim Nicholas Taleb, Skin in the game. "Ergodicity is not statistically identifiable, not observable, and there is no test for time series that gives ergodicity, similar to Dickey-Fuller for stationarity (or Phillips-Perron for integration order)." "If your result is obtained from the observation of a time series, how can you make claims about the ensemble probability measure?"


  1. Table: Important Discrete Random Variables
  2. Table: Important Continuous Random Variables

Probabilistic Models

  • Reference Distribution
  • Distributions from Discrete Random Process
  • Distributions from Continuous Random Process
  • Asymptotic Distributions
  • Gaussian-related Distributions

Fourier Transforms, Z-transform

Table 1: Interpretations of probability

Ontic (实存) probability Epistemic (认识) probability
long run frequency objective degree of belief
single-case propensity subjective degree of belief

🏷 Category=Probability