- Probability space
- Conditional probability
- Total probability equation, Bayes formular
- 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
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.
(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?"
- Table: Important Discrete Random Variables
- Table: Important Continuous Random Variables
- 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
||subjective degree of belief