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

- Probability space;
- Conditional probability;
- Total probability equation, Bayes formular;
- Independence;

- Measurable function on probability space;
- Distribution: CDF, PDF, PMF;
- Integration (Expectation), Riemann-Stieltjes integral;
- Characteristic function and moments
- Function Analytic Approach to Probability;

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

- Conditional expectation;
- Hierarchical models;
- Independence as an assumption and simplification;
- Covariance and correlation;

Special topics:

- Inequalities and Identities;
- Gaussian Random Variables;
- Transformation and Decomposition;
- Statistics of Extremes;

Note that mathematical theories go complicated very fast. In fact, the description of a random process already requires too much information, which is impractical. To put the mathematical model into practical use, vast simplification is needed.

Stochastic analysis:

- Stochastic Convergence;
- Limit Theorems;
- Random Process Analysis: convergence; continuity; differentiability; integrability;
- Spectral Representation: K-L expansion; PCE;

(Ref: Scholtz, P266.)

Random processes:

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?"

Fourier Transforms, Z-transform:

Table: **Fourier Transform** and Its Inverse Transform as the Integral Transforms Defined by Kernels.
(In the form of $a e^{bixy}$, where $a$ is normalization and $b$ is oscillatory factor.)

Convention | Forward kernel | Inverse kernel |
---|---|---|

physics | $e^{-ixy}$ | $(2\pi)^{-1}e^{ixy}$ |

physics, symmetric | $(2\pi)^{-1/2} e^{-ixy}$ | $(2\pi)^{-1/2} e^{ixy}$ |

mathematics | $e^{-2\pi ixy}$ | $e^{2\pi ixy}$ |

general | $│b│^{1/2}(2\pi)^{(-1+a)/2}e^{-ibxy}$ | $│b│^{1/2}(2\pi)^{(-1-a)/2}e^{ibxy}$ |

**Sinc function** $\text{sinc}(x) = \frac{\sin x}{x}$, $x \ne 0$.
**Normalized sinc function** $\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$, $x \ne 0$,
because its integral on the real line equals one:
$\int_{-\infty}^\infty \frac{\sin(\pi x)}{\pi x} = 1$.
**Rectangular function** is the centered unit-length indicator function:
$\text{rect}(x) = 1(|x| \le 1/2)$.
The Fourier transform (in math convention) of the normalized sinc / rectagular function
is the rectagular / normalized sinc function.
The Fourier transform (in physics convention) of the sinc function
is the constant function that equals pi in between -1 and 1:
$\int_{-\infty}^\infty e^{-ixt} \frac{\sin x}{x} = \pi \cdot 1(|t| \le 1)$.

Table: 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 |