## 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.)

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

- Table: Important Discrete Random Variables
- 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