Probability is interchangeable with proportion; although one is theoretical, the other arithmetical.

## 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

## 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.

### Random Processes (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?"

## Appendices

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