Nguồn ngoài

Probability, Mathematical Statistics, and Stochastic Processes

This text is devoted to probability, mathematical statistics, and stochastic processes, and is intended for teachers and students of these subjects. The site consists of an integrated set of components that includes expository text, interactive web apps, data sets, biographical sketches, and an object library.

Probability Theory Statistics

Tác giả: Kyle Siegrist
Nhà xuất bản: University of Alabama in huntsville
Nguồn: LibreTexts
Giấy phép: CC BY
Ngôn ngữ: EN
Danh mục: Khoa học cơ bản

Đọc trực tuyến Tải PDF (nguồn mở)

Mục lục· 211 mục

01: Foundations
1.01: Sets
1.02: Functions
1.03: Relations
1.04: Partial Orders
1.05: Equivalence Relations
1.06: Cardinality
1.07: Counting Measure
1.08: Combinatorial Structures
1.09: Topological Spaces
1.10: Metric Spaces
1.11: Measurable Spaces
1.12: Special Set Structures
02: Probability Spaces
2.01: Random Experiments
2.02: Events and Random Variables
2.03: Probability Measures
2.04: Conditional Probability
2.05: Independence
2.06: Convergence
2.07: Measure Spaces
2.08: Existence and Uniqueness
2.09: Probability Spaces Revisited
2.10: Stochastic Processes
2.11: Filtrations and Stopping Times
03: Distributions
3.01: Discrete Distributions
3.02: Continuous Distributions
3.03: Mixed Distributions
3.04: Joint Distributions
3.05: Conditional Distributions
3.06: Distribution and Quantile Functions
3.07: Transformations of Random Variables
3.08: Convergence in Distribution
3.09: General Distribution Functions
3.10: The Integral With Respect to a Measure
3.11: Properties of the Integral
3.12: General Measures
3.13: Absolute Continuity and Density Functions
3.14: Function Spaces
04: Expected Value
4.01: Definitions and Basic Properties
4.02: Additional Properties
4.03: Variance
4.04: Skewness and Kurtosis
4.05: Covariance and Correlation
4.06: Generating Functions
4.07: Conditional Expected Value
4.08: Expected Value and Covariance Matrices
4.09: Expected Value as an Integral
4.10: Conditional Expected Value Revisited
4.11: Vector Spaces of Random Variables
4.12: Uniformly Integrable Variables
4.13: Kernels and Operators
05: Special Distributions
5.01: Location-Scale Families
5.02: General Exponential Families
5.03: Stable Distributions
5.04: Infinitely Divisible Distributions
5.05: Power Series Distributions
5.06: The Normal Distribution
5.07: The Multivariate Normal Distribution
5.08: The Gamma Distribution
5.09: Chi-Square and Related Distribution
5.10: The Student t Distribution
5.11: The F Distribution
5.12: The Lognormal Distribution
5.13: The Folded Normal Distribution
5.14: The Rayleigh Distribution
5.15: The Maxwell Distribution
5.16: The Levy Distribution
5.17: The Beta Distribution
5.18: The Beta Prime Distribution
5.19: The Arcsine Distribution
5.20: General Uniform Distributions
5.21: The Uniform Distribution on an Interval
5.22: Discrete Uniform Distributions
5.23: The Semicircle Distribution
5.24: The Triangle Distribution
5.25: The Irwin-Hall Distribution
5.26: The U-Power Distribution
5.27: The Sine Distribution
5.28: The Laplace Distribution
5.29: The Logistic Distribution
5.30: The Extreme Value Distribution
5.31: The Hyperbolic Secant Distribution
5.32: The Cauchy Distribution
5.33: The Exponential-Logarithmic Distribution
5.34: The Gompertz Distribution
5.35: The Log-Logistic Distribution
5.36: The Pareto Distribution
5.37: The Wald Distribution
5.38: The Weibull Distribution
5.39: Benford's Law
5.40: The Zeta Distribution
5.41: The Logarithmic Series Distribution
06: Random Samples
6.01: Introduction
6.02: The Sample Mean
6.03: The Law of Large Numbers
6.04: The Central Limit Theorem
6.05: The Sample Variance
6.06: Order Statistics
6.07: Sample Correlation and Regression
6.08: Special Properties of Normal Samples
07: Point Estimation
7.01: Estimators
7.02: The Method of Moments
7.03: Maximum Likelihood
7.04: Bayesian Estimation
7.05: Best Unbiased Estimators
7.06: Sufficient Complete and Ancillary Statistics
08: Set Estimation
8.01: Introduction to Set Estimation
8.02: Estimation the Normal Model
8.03: Estimation in the Bernoulli Model
8.04: Estimation in the Two-Sample Normal Model
8.05: Bayesian Set Estimation
09: Hypothesis Testing
9.01: Introduction to Hypothesis Testing
9.02: Tests in the Normal Model
9.03: Tests in the Bernoulli Model
9.04: Tests in the Two-Sample Normal Model
9.05: Likelihood Ratio Tests
9.06: Chi-Square Tests
10: Geometric Models
10.01: Buffon's Problems
10.02: Bertrand's Paradox
10.03: Random Triangles
11: Bernoulli Trials
11.01: Introduction to Bernoulli Trials
11.02: The Binomial Distribution
11.03: The Geometric Distribution
11.04: The Negative Binomial Distribution
11.05: The Multinomial Distribution
11.06: The Simple Random Walk
11.07: The Beta-Bernoulli Process
12: Finite Sampling Models
12.01: Introduction to Finite Sampling Models
12.02: The Hypergeometric Distribution
12.03: The Multivariate Hypergeometric Distribution
12.04: Order Statistics
12.05: The Matching Problem
12.06: The Birthday Problem
12.07: The Coupon Collector Problem
12.08: Polya's Urn Process
12.09: The Secretary Problem
13: Games of Chance
13.01: Introduction to Games of Chance
13.02: Poker
13.03: Simple Dice Games
13.04: Craps
13.05: Roulette
13.06: The Monty Hall Problem
13.07: Lotteries
13.08: The Red and Black Game
13.09: Timid Play
13.10: Bold Play
13.11: Optimal Strategies
14: The Poisson Process
14.01: Introduction to the Poisson Process
14.02: The Exponential Distribution
14.03: The Gamma Distribution
14.04: The Poisson Distribution
14.05: Thinning and Superpositon
14.06: Non-homogeneous Poisson Processes
14.07: Compound Poisson Processes
14.08: Poisson Processes on General Spaces
15: Renewal Processes
15.01: Introduction
15.02: Renewal Equations
15.03: Renewal Limit Theorems
15.04: Delayed Renewal Processes
15.05: Alternating Renewal Processes
15.06: Renewal Reward Processes
16: Markov Processes
16.01: Introduction to Markov Processes
16.02: Potentials and Generators for General Markov Processes
16.03: Introduction to Discrete-Time Chains
16.04: Transience and Recurrence for Discrete-Time Chains
16.05: Periodicity of Discrete-Time Chains
16.06: Stationary and Limiting Distributions of Discrete-Time Chains
16.07: Time Reversal in Discrete-Time Chains
16.08: The Ehrenfest Chains
16.09: The Bernoulli-Laplace Chain
16.10: Discrete-Time Reliability Chains
16.11: Discrete-Time Branching Chain
16.12: Discrete-Time Queuing Chains
16.13: Discrete-Time Birth-Death Chains
16.14: Random Walks on Graphs
16.15: Introduction to Continuous-Time Markov Chains
16.16: Transition Matrices and Generators of Continuous-Time Chains
16.17: Potential Matrices
16.18: Stationary and Limting Distributions of Continuous-Time Chains
16.19: Time Reversal in Continuous-Time Chains
16.20: Chains Subordinate to the Poisson Process
16.21: Continuous-Time Birth-Death Chains
16.22: Continuous-Time Queuing Chains
16.23: Continuous-Time Branching Chains
17: Martingales
17.01: Introduction to Martingalges
17.02: Properties and Constructions
17.03: Stopping Times
17.04: Inequalities
17.05: Convergence
17.06: Backwards Martingales
18: Brownian Motion
18.01: Standard Brownian Motion
18.02: Brownian Motion with Drift and Scaling
18.03: The Brownian Bridge
18.04: Geometric Brownian Motion