Probability

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

• Probability Notes
• Scanned Notes on Probability Theory

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

• Inequalities and Identities
• Gaussian Random Variables
• Transformation and Decomposition
• Statistics of Extremes

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.

• Midterm summary
• Miscellaneous
• Markov Chain

Stochastic Analysis

• Stochastic Convergence
• Limit Theorems
• Random Process Analysis
  • Convergence
  • Continuity
  • Differentiability
  • Integrability
• Spectral Representation
  • K-L expansion
  • PCE

Random Processes

(Ref: Scholtz, P266.)

• Random Process
  • w.s.s.-LTI relationship
  • Poisson Process
  • Ergodic Process
• System
• Simulation
• Estimation

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

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

• Transforms
• Inversion of Fourier Transform by Residues

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 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

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🏷 Category=Probability