Probability. Sample Space. Discrete and Continuous Random Variables. Discrete and Continuous Probability Distributions. Central Limit Theorem. Chebyshev’s Theorem. Mathematical Expectation and Variances. Moment Generating Functions.Estimation of Parameters. Hypothesis Testing for one and two samples.
MA2510 - EF3250
Also : MIT Intro to Prob and stats 18.05 ( No video )

Stanford Probability Theory: STAT310/MATH230 No video

Advanced Stochastic Processes: No video

Prob Stats for Eng and Sci
https://www.youtube.com/watch?v=rmGDNf3TKQE&list=PLuLrbLfHQopo7plRJZ8gkKZ6sdhOv3Q4B&index=1

Probability foundations

• Textbook
  • 2 Probability: Sample Space; Events; Counting Sample Points; Probability of an Event; Additive Rules; Conditional Probability; Multiplicative Rules; Bayes’ Rule; Exercises and Review Exercises[1][2]
• MIT
  • 1 Probability Models and Axioms; 2 Conditioning and Bayes’ Rule; 3 Independence; 4 Counting[2][1]
• Stanford
  • 16 概率的解释; 17 互补事件、等可能结果的加法和乘法; 18 四则法则：如何处理至少一个的问题; 19 通过完全列举解决问题; 20 贝叶斯法则; 21 贝叶斯分析[1][2]
• Harvard
  • Probability: Counting; Random variables, distributions, quantiles, mean variance; Conditional probability, Bayes’ theorem, base rate fallacy; Joint distributions, covariance, correlation, independence; Central limit theorem[2][1]

Discrete random variables and distributions

• Textbook
  • 3.2 Discrete Probability Distributions; 5 Some Discrete Probability Distributions: Discrete Uniform; Binomial and Multinomial; Hypergeometric; Negative Binomial and Geometric; Poisson Distribution and Poisson Process; Exercises/Review[1][2]
• MIT
  • 5 Discrete Random Variables; PMFs; Expectations; 6 Discrete RV Examples; Joint PMFs; 7 Multiple Discrete RVs: Expectations, Conditioning, Independence; 13 Bernoulli Process; 14–15 Poisson Process I–II[2][1]
• Stanford
  • 29 二项分布和二项系数; 30 二项式公式; 31 随机变量和概率直方图; 32 无放回的二项抽样的正态近似; 24 正态曲线 (as bridge to continuous); 25 经验法则[1][2]
• Harvard
  • Probability section covers random variables and distributions; later Statistics II includes resampling and regression (discrete named less explicitly)[2][1]

Continuous random variables and key continuous families

• Textbook
  • 3.3 Continuous Probability Distributions; 6 Some Continuous Probability Distributions: Continuous Uniform; Normal; Gamma and Exponential; Chi-Squared; Lognormal; Weibull (Optional); Areas under Normal; Applications; Normal Approximation to the Binomial; Exercises/Review[1][2]
• MIT
  • 8 Continuous Random Variables; 9 Multiple Continuous Random Variables; 10 Continuous Bayes’ Rule; Derived Distributions; 11 Derived Distributions; Convolution; Covariance and Correlation[2][1]
• Stanford
  • 24 正态曲线; 25 经验法则; 26 标准化与标准正态; 27 正态近似; 28 用正态近似计算百分位数; 49 给定 x 的正态近似[1][2]
• Harvard
  • Probability: random variables, distributions, quantiles, mean, variance; joint distributions, covariance, correlation[2][1]

Joint distributions, covariance, correlation, independence

• Textbook
  • 3.4 Joint Probability Distributions; 4.2 Variance and Covariance; 4.3 Linear Combinations; 3.5 Potential Misconceptions[1][2]
• MIT
  • 6 Joint PMFs; 7 Multiple Discrete RVs (independence, conditioning); 9 Multiple Continuous RVs; 11 Covariance and Correlation[2][1]
• Stanford
  • 44 相关系数; 45 相关性测量线性关联[1][2]
• Harvard
  • Joint distributions, covariance, correlation, independence (explicit bullet)[2][1]

Mathematical expectation and moment tools

• Textbook
  • 4 Mathematical Expectation: Mean; Variance and Covariance; Means/Variances of Linear Combinations; Chebyshev’s Theorem; Moments/MGFs in 7.3; Exercises/Review[1][2]
• MIT
  • 5 Expectations for discrete RVs; 7 Expectations for multiple RVs; 11 Convolution; 12 Iterated Expectations[2][1]
• Stanford
  • 35 期望值和标准误差; 36 总和百分比的期望值和标准误差以及模拟计算; 37 平方根定律[1][2]
• Harvard
  • Covered across Probability and Statistics I/II where expectations underlie inference[2][1]

Transformations, MGFs, functions of RVs

• Textbook
  • 7 Functions of Random Variables (Optional): Transformations; Moments and MGFs[1][2]
• MIT
  • 10–11 Derived distributions; convolution; MGFs implied through moments; 12 Iterated Expectations[2][1]
• Stanford
  • 49 给定 x 的正态近似; 50 残差图、异方差性和变换 (transforms in regression context)[1][2]
• Harvard
  • Not explicit; implicit under Probability and regression portions[2][1]

Sampling distributions and limit theorems

• Textbook
  • 8 Fundamental Sampling Distributions: Random Sampling; Important Statistics; Sampling Distributions; Sampling Distribution of Means; of S; t-Distribution; F-Distribution; Review[1][2]
• MIT
  • 19 Weak Law of Large Numbers; 20 Central Limit Theorem[2][1]
• Stanford
  • 38 抽样分布; 40 大数定律; 41 中心极限定理; 42 适用条件; 54 利用中心极限定理计算置信区间[1][2]
• Harvard
  • Probability bullet ends with Central limit theorem; Statistics II builds on sampling/CI/testing[2][1]

Estimation: point, intervals, standard errors

• Textbook
  • 9 Estimation Problems: Classical Methods; Estimating Means, Proportions, Variances; Standard Error; Prediction Intervals; Tolerance Limits; MLE (Optional); Review[1][2]
• MIT
  • 21–22 Bayesian Statistical Inference I–II; 23–25 Classical Statistical Inference I–III[2][1]
• Stanford
  • 34 参数和统计量; 35 标准误差; 53 置信区间的解释; 54 基于 CLT 的置信区间; 55 用自助法估计标准误差; 56 更多关于置信区间; 69 参数自助法和自助法置信区间[1][2]
• Harvard
  • Statistics I: Bayesian inference with known priors, probability intervals, conjugate priors; Statistics II: Bayesian with unknown priors; frequentist CIs[2][1]

Hypothesis testing

• Textbook
  • 10 One- and Two-Sample Tests: Concepts; Testing; One-/Two-Tailed; p-values; Tests on Means (known/unknown variance); Two-sample means; Sample size; Graphical comparison; Proportion tests; Variance tests; Goodness-of-fit; Independence; Homogeneity; Several proportions; Case study; Review[1][2]
• MIT
  • 23–25 Classical inference series cover hypothesis tests and intervals[2][1]
• Stanford
  • 57 假设检验的思想; 58 建立检验统计量; 59 P 值; 61 t 检验; 63 双样本 z 检验; 64 配对样本; 73 卡方检验（均质性与独立性）; 74 比较多个均值; 75–77 方差分析与 F 分布; 80 Bonferroni 修正、虚发现率和数据拆分[1][2]
• Harvard
  • Statistics II: Frequentist significance tests and confidence intervals[2][1]

Regression and correlation

• Textbook
  • Appears via covariance, correlation, graphical methods, but no full regression chapter in the provided slice; related in 8.3 data displays, 10.10 graphical methods for comparing means[1][2]
• MIT
  • 11 Covariance and Correlation; regression as application is not a separate lecture in 6.041[2][1]
• Stanford
  • 44 相关系数; 45 线性关联; 46 回归线与最小二乘; 47 回归到均值与谬误; 48 根据 x 预测 y 与反向; 50 残差图、异方差性和变换; 70 回归中的自助法[1][2]
• Harvard
  • Statistics II includes Linear regression explicitly[2][1]

Resampling and multiple testing

• Textbook
  • Not in provided chapters; MLE optional is included[1][2]
• MIT
  • Not core in 6.041 lecture list; focus is classical/Bayesian foundations[2][1]
• Stanford
  • 55 自助法原理估计标准误差; 69 自助法置信区间; 70 回归中的自助法; 79 数据窥探、多重检验、可重复性与可复制性; 80 Bonferroni 修正、虚发现率和数据拆分[1][2]
• Harvard
  • Statistics II: Resampling methods: bootstrapping[2][1]

Graphical methods and EDA

• Textbook
  • 8.3 Data Displays and Graphical Methods[1][2]
• MIT
  • Not a distinct lecture; visuals used but not a unit[2][1]
• Stanford
  • 3 饼图、条形图和直方图; 4 箱线图和散点图; 39 三个直方图; 50 残差图[1][2]
• Harvard
  • Implicit within statistics content, not a separate bullet[2][1]

Experimental design and sampling basics

• Textbook
  • 8.1 Random Sampling; 8.2 Important Statistics; 8.4–8.8 Sampling distributions including t and F[1][2]
• MIT
  • Sampling is implicit in inference and CLT lectures[2][1]
• Stanford
  • 11 简单随机抽样与分层抽样; 12 偏差和偶然误差; 13 观察与实验、混淆与安慰剂; 14 随机对照实验逻辑[1][2]
• Harvard
  • Framed across Statistics I/II under inference setup[2][1]

Stochastic processes (optional/enrichment)

• Textbook
  • 5.6 Poisson Process within discrete distributions; no Markov chains in provided slice[1][2]
• MIT
  • 13 Bernoulli Process; 14–15 Poisson Process I–II; 16–18 Markov Chains I–III[2][1]
• Stanford
  • Not covered as a named module; some probability process ideas appear implicitly[1][2]
• Harvard
  • Not within listed bullets; focus on inference/regression[2][1]

Unified syllabus sequencing note

• Start with Probability foundations, then Discrete and Continuous RVs, Joint/Covariance/Correlation, Expectation and transforms, Sampling distributions and CLT, Estimation (Bayesian then Classical), Hypothesis testing, Regression and correlation, Resampling and multiple testing, plus Process topics as enrichment. This ordering respects Textbook and MIT flow, while incorporating Stanford’s EDA, regression, and resampling modules and Harvard’s Bayesian-first framing under Statistics I/II. Every chapter/lecture/item listed above is retained under its cluster so none are omitted.[1][2]

Derivations

Applied Derivation - In Probabilistic, Statistics and Finance

Main reference

1. Probability and Statistics for engineers and scientists
2. MIT
   

Open Courses

https://www.reddit.com/r/statistics/comments/1gk3iqf/e_best_video_series_on_probability_and_statistics/

StatQuest by Josh Starmer

MIT 6.041 -> Also see18.650

• https://ocw.mit.edu/courses/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/resources/lecture-1-probability-models-and-axioms/
• http://youtube.com/watch?v=VPZD_aij8H0
  MIT 6.041 Probabilistic Systems Analysis Summaries
  Also see the recitations of the courses " https://ocw.mit.edu/courses/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/pages/recitations/"

Stanford
(free) https://www.bilibili.com/video/BV185411k7mr/
(paid original) https://www.coursera.org/learn/stanford-statistics?action=enroll
Stanford Introduction to Statistics Summaries

• 概率論與數理統計完整版/梁恒清華大學公開課：本課程將引導學生學習和理解定量描述隨機性的基本數學模型和理論方法。（來源：YouTube）
• 《統計學習方法》（李航）-北京大學：全面系統地介紹了統計學習的主要方法，特別是監督學習方法。（來源：Bilibili）
• 《看見統計》 （ https://seeing-theory.brown.edu/cn.html ）

Harvard STAT110x (Story telling)

• https://learning.edx.org/course/course-v1:HarvardX+STAT110x+2T2025/block-v1:HarvardX+STAT110x+2T2025+type@sequential+block@78ddcb79a5634fa0acb1bf492579238e/block-v1:HarvardX+STAT110x+2T2025+type@vertical+block@1dcb275ff05342a7abc745873c6c1787

Problems

Problem log - MA2510

How Probability and Statistics lay foundations for Stochastic Calculus

MA2510 - EF3250

Some Course-oriented materials

MA2510_high_score_resources_zh

Important intro

• 統計學架構：統計學的架構大概長這樣：樞紐量、母體→樣本→統計量、檢定量。而統計大概可以分成兩大區：機率論、統計推論。（來源：PTT）

数学专业 （本科，研究生）       一般人们对概率论这门学科的理解可以划分为三个层次：
1古典型，未受过任何相关训练的人都属于此类，只能够理解一些离散的（古典的）概率模型；
2近代型，通常指学过概率论基础的，从微积分的角度理解各种连续分布，概率模型的数字特征；
3现代型，抽象地从测度论和实分析高度理解，建立在测度基础上的概率论通常所谓的高等概率论。
           选一本适合自己的好的教材对自己以后的学习是决定性的重要--这是学数学的人首先必须明白的--不仅是对概率方向，对数学的各个分支都是如此。大一的时候齐名友老师跟我特别提到过这一点，可惜我当时不以为然，结果走了很多弯路，到研究生以后才慢慢明白这个道理。 一本山寨小学校的老师七拼八凑编写的烂书，常常对学习（特别是自学）不仅无益反而有害，因为你往往浪费了时间却只能得到这个一些支离破碎的印象，这样你会遗忘得很快，很可能到头来你还得重新学一遍 ；另一些时候，你选择了众人推荐的名著，但你如果当前的水平达不到一定的层次，它往往会打击你的信心让你灰心丧气，甚至会让你不再有学下去的欲望。这两种情形显然都是人们应该尽量避免的。
       需要指出的是，有的书适合作教材，有的书却只适合作参考书；就算都是教材，它定位的读者群体也可能不一样。每个人都应该根据自己的实际情况做出选择。一般好书大多都是国外的，所以如果有可能最好去看国外的原版书，就算没有这个能力也应该去锻炼这个能力。
       读原版书其实没看起来的那么难，你不需要懂得任何高深的语法，记熟100个单词/词组就能轻易上手，
       记熟300个你就能在大多数情况下不需要字典了。我记得我法语学了不到一年就来到法国读书，老师上课基本听不懂，只能自己找书看，而图书馆里绝大多数参考书都是法语的（当时不知道在网上找书）。按说我当时法语应该比大多数中国大学生英语要远差，但我抱着一本法语的拓扑书回家一边查字典一边看，两三天就完全适应了。真正看外文原版书，要克服的首要困难永远都是数学本身，而不是生词或者语法。
 
 我推荐的学习方法是这样的：读一本简单而直观的入门书，
 这样能比较容易地把握一个领域的主干，
 明白它要达到哪些目的，
 通过什么样的方法，
 关键性的定理有哪些；
 等掌握大体框架之后再找一本详尽而严密的教材慢慢推敲其细节。
 中文的书我没什么好推荐的--在国内的时候看的书质量都不高（当时抱着一本书就看，对好书和烂书也没有概念）而出国之后就没再看过中文书了。我依稀记得汪嘉冈的《现代概率基础》还不错，其它的我就不知道了。对于外文书，我倒是有很多可以推荐。这样我首先要推荐的是David Williams写的Probability with martingales。书写得很薄，严格意义上说它不是一本教材，但完全可以把它当做现代概率论和鞅理论的入门书来看。我觉得很少有书能够写得象它那样把严密性，直观性以及趣味性完美的融合到一起，并且自成体系(即所谓self-contained，就是说你不需要一边看这本书一边在别的书里寻找相关定理，定义或者其它背景知识)。它只引入对主题有帮助的概念，因此这样读者就可以不必顾及细枝末节从而能够快速领悟其精髓。等你入门之后，可以看的进阶级书就很多了，比如Chung Kai Lai的A course in probability theory。