概率论与数理统计导论
《麻省理工 MIT 6.041 概率论与数理统计导论》课程提供了一个全面的概率与统计学基础，旨在帮助学生理解随机现象的数学模型，并培养运用概率理论分析实际问题的能力。课程开始时介绍了概率的基本公理与模型，通过各种经典的概率谜题，如蒙提霍尔问题、抛硬币实验和随机游走问题，引导学生理解条件概率、独立性和贝叶斯定理等核心概念。学生还将学习如何计算离散和连续随机变量的概率分布、期望与方差，并深入探讨多维随机变量及其应用。
随着课程的深入，学生将接触到更复杂的概率过程，如泊松过程、马尔可夫链和伯努利过程，以及如何运用这些工具建模和解决实际问题。此外，课程还介绍了重要的统计推断方法，包括经典统计推断、贝叶斯统计推断和参数推断，帮助学生通过数据分析得出合理的结论。通过对大数法则、中心极限定理和其他概率界限的学习，学生将理解随机现象的长期行为，并能够应用这些理论进行各种统计推断。
总体而言，这门课程不仅为学生提供了概率论和数理统计的扎实基础，还通过一系列实际案例和应用，帮助学生将理论与实践结合，掌握如何在不确定性下进行决策和推理。这些工具在工程、经济、计算机科学等领域都有广泛的应用。
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1.概率模型与公理-Probability Models and Axioms
2.两事件差的概率-The Probability of the Difference of Two Events
3.天才与巧克力问题-Geniuses and Chocolates
4.正方形上的均匀分布-Uniform Probabilities on a Square
5.条件概率与贝叶斯定理-Conditioning and Bayes' Rule
6.抛硬币谜题-A Coin Tossing Puzzle
7.条件概率案例-Conditional Probability Example
8.蒙提霍尔问题（三门问题）-The Monty Hall Problem
9.独立性-Independence
10.随机游走者-Random Walker
11.噪声信道通信-Communication over a Noisy Channel
12.网络可靠性-Network Reliability
13.国际象棋锦标赛问题-Chess Tournament Problem
14.计数原理-Counting
15.棋盘上的车-Rooks on a Chessboard
16.超几何分布-Hypergeometric Probabilities
17.离散随机变量I-Discrete Random Variables I
18.公交乘客抽样-Sampling People on Buses
19.随机变量函数的概率质量函数-PMF of a Function of a Random Variable
20.离散随机变量II-Discrete Random Variables II
21.随机次数的抛硬币实验-Flipping a Coin a Random Number of Times
22.联合概率质量函数（PMF）练习1-Joint Probability Mass Function (PMF) Drill 1
23.赠券收集问题-The Coupon Collector Problem
24.离散随机变量III-Discrete Random Variables III
25.联合概率质量函数（PMF）练习2-Joint Probability Mass Function (PMF) Drill 2
26.连续随机变量-Continuous Random Variables
27.累积分布函数（CDF）计算-Calculating a Cumulative Distribution Function (CDF)
28.混合分布案例-A Mixed Distribution Example
29.指数分布的均值与方差-Mean & Variance of the Exponential
30.正态分布概率计算-Normal Probability Calculation
31.多维连续随机变量-Multiple Continuous Random Variables
32.三角形上的均匀分布-Uniform Probabilities on a Triangle
33.三线段构成三角形的概率-Probability that Three Pieces Form a Triangle
34.心不在焉的教授问题-The Absent Minded Professor
35.连续贝叶斯定理与推导分布-Continuous Bayes' Rule; Derived Distributions
36.从连续测量推断离散随机变量-Inferring a Discrete Random Variable from a Continuous Measurement
37.从离散测量推断连续随机变量-Inferring a Continuous Random Variable from a Discrete Measurement
38.推导分布案例-A Derived Distribution Example
39.[X]的概率密度函数（PDF）-The Probability Distribution Function (PDF) of [X]
40.救护车行驶时间问题-Ambulance Travel Time
41.推导分布（续）与协方差-Derived Distributions (ctd.); Covariance
42.两个独立指数随机变量之差-The Difference of Two Independent Exponential Random Variables
43.离散与连续随机变量之和-The Sum of Discrete and Continuous Random Variables
44.迭代期望-Iterated Expectations
45.折棒问题中的方差-The Variance in the Stick Breaking Problem
46.货箱与零件问题-Widgets and Crates
47.条件期望与方差的应用-Using the Conditional Expectation and Variance
48.随机次数的抛硬币实验-A Random Number of Coin Flips
49.随机偏差的硬币问题-A Coin with Random Bias
50.伯努利过程-Bernoulli Process
51.伯努利过程练习-Bernoulli Process Practice
52.泊松过程I-Poisson Process I
53.竞争指数分布-Competing Exponentials
54.泊松过程II-Poisson Process II
55.爱尔朗到达下的随机入射-Random Incidence Under Erlang Arrivals
56.马尔可夫链I-Markov Chains I
57.构建马尔可夫链-Setting Up a Markov Chain
58.马尔可夫链练习1-Markov Chain Practice 1
59.马尔可夫链II-Markov Chains II
60.马尔可夫链III-Markov Chains III
61.首次通过均值与递归时间-Mean First Passage and Recurrence Times
62.弱大数定律-Weak Law of Large Numbers
63.依概率收敛与均方收敛（上）-Convergence in Probability and in the Mean Part 1
64.依概率收敛与均方收敛（下）-Convergence in Probability and in the Mean Part 2
65.依概率收敛案例-Convergence in Probability Example
66.中心极限定理-Central Limit Theorem
67.概率界限-Probabilty Bounds
68.中心极限定理应用-Using the Central Limit Theorem
69.贝叶斯统计推断I-Bayesian Statistical Inference I
70.贝叶斯统计推断II-Bayesian Statistical Inference II
71.均匀分布参数推断（上）-Inferring a Parameter of Uniform Part 1
72.均匀分布参数推断（下）-Inferring a Parameter of Uniform Part 2
73.统计推断案例-An Inference Example
74.经典统计推断I-Classical Statistical Inference I
75.经典统计推断II-Classical Inference II
76.经典统计推断III-Classical Inference III