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15.085J 概率论基础

基于测度论（Measure-theoretic）的严谨概率论，覆盖 sigma 代数、测度与积分。 依概率收敛、几乎处处收敛、Lp 收敛与依分布弱收敛四大收敛性层级关系的严格证明。 特征函数、大数定律、中心极限定理与条件期望作为正交投影的代数几何解析。

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15.085J 概率论基础

MIT Course 15 · Operations Research/Statistics · 课号 15.085J

TL;DR

  • 基于测度论（Measure-theoretic）的严谨概率论，覆盖 sigma 代数、测度与积分。
  • 依概率收敛、几乎处处收敛、Lp 收敛与依分布弱收敛四大收敛性层级关系的严格证明。
  • 特征函数、大数定律、中心极限定理与条件期望作为正交投影的代数几何解析。

课程简介 / Description

This is a course on the fundamentals of probability geared towards first or second-year graduate students who are interested in a rigorous development of the subject. The course covers sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, and limit theorems. There is also a number of additional topics such as: language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; and deeper understanding of conditional distributions and expectations.

讲义 / Lecture Notes

  • 1
    • Probabilistic Models and Probability Measures (PDF)
  • 2
    • Two Fundamental Probabalistic Models (PDF)
  • 3
    • Conditioning and Independence (PDF).
  • 4
    • Random Variables (PDF)
  • 5
    • Discrete Random Variables and Their Expectations (PDF)
  • 6
    • More on Discrete Random Variables and Their Expectations (PDF)
  • 7
    • Abstract Integration I (PDF)
  • 8
    • Abstract Integration II (PDF)
  • 9
    • Product Measure and Fubini's Theorem (PDF)
  • 10
    • Continous Random Variables I (PDF)
  • 11
    • Continous Random Variables II (PDF)
  • 12
    • Derived Distributions (PDF)
  • 13
    • Moment Generating Functions (PDF)
  • 14
    • Multivariate Normal Distributions (PDF)
  • 15
    • Multivariate Normal Distributions. Characteristic Functions (PDF)
  • 16
    • Convergence of Random Variables (PDF)
  • 17
    • Laws of Large Numbers I (PDF)
  • 18
    • Laws of Large Numbers II (PDF)
  • 19
    • Uniform Integrability. Convergence of Series (PDF)
  • 20
    • The Basics of Stochastic Processes (PDF)
  • 21
    • Markov Chains I (PDF)
  • 22
    • Markov Chains II (PDF)
  • 23
    • Markov Chains III (PDF)
  • 24
    • Infinite Markov Chains. Continuous Time Markov Chains (PDF)
  • 25
    • Martingales I (PDF)
  • 26
    • Martingales II (PDF)

本资料整理自 MIT OpenCourseWare，遵循 Creative Commons BY-NC-SA 4.0 许可协议，仅供个人学习使用。