├── Additional Materials ├── Extra Material │ ├── Bayesian Infernece.pptx │ ├── Bayes’+Theorem.jpg │ ├── Convoulation Continous Problems PART-B.pdf │ ├── Exponetial memorylesnees property.pdf │ ├── Probability Cheatbook.pptx │ ├── Standard Table of value Z.pdf │ ├── Textbook_Tables (1).pdf │ ├── UW_MATH-STAT395_functions-random-variables.pdf │ ├── bayesian - What's diff bet a confidence interval and credible interval.pdf │ ├── conditional-expectation__110.png │ ├── log transformation minimize and maximize.png │ ├── possion process paramter for merging part2.PNG │ ├── possion process paramter for merging.PNG │ ├── probability - Find the covariances of a multinomial distribution.pdf │ ├── probability - Sum of a random number of r.v.'s - Cross Validated.pdf │ └── probability distributions - What is the pdf of $Y = _log X$_.pdf ├── Lecture Slide Material │ ├── lectureslides_L01annotatedslides.pdf │ ├── lectureslides_L02annotatedslides.pdf │ ├── lectureslides_L03annotatedslides.pdf │ ├── lectureslides_L04-annotated-slides.pdf │ ├── lectureslides_L05-annotated-slides.pdf │ ├── lectureslides_L06-annotated-slides.pdf │ ├── lectureslides_L07-annotated-slides.pdf │ ├── lectureslides_L08-annotated-slides.pdf │ ├── lectureslides_L09-annotated-slides.pdf │ ├── lectureslides_L10-annotated-slides.pdf │ ├── lectureslides_L11-annotated-slides.pdf │ ├── lectureslides_L12-annotated-slides.pdf │ ├── lectureslides_L13-annotate-dslides.pdf │ ├── lectureslides_L14-annotated.pdf │ ├── lectureslides_L15-annotated.pdf │ ├── lectureslides_L16-annotated.pdf │ ├── lectureslides_L17-annotated.pdf │ ├── lectureslides_L18-annotated-slides.pdf │ ├── lectureslides_L19-annotated-slides.pdf │ ├── lectureslides_L20-annotated-slides.pdf │ ├── lectureslides_L21-annotated-slides.pdf │ ├── lectureslides_L22-annotated-slides.pdf │ ├── lectureslides_L23-annotated-slides.pdf │ ├── lectureslides_L24-annotated-slides.pdf │ ├── lectureslides_L25-annotated-slides.pdf │ └── lectureslides_L26-annotated-slides.pdf ├── Unit 1 & 2 Revise │ ├── Demorgan Law.PNG │ ├── Independece Basic concept │ │ ├── Important example of two event parwise and umpairwise independent.PNG │ │ ├── Independent example concrete.PNG │ │ ├── Independent may affect from dependent intivetive sense.png │ │ ├── Independent possibilties.PNG │ │ ├── Not in Independent.PNG │ │ ├── independent event fact.PNG │ │ └── probability - Could someone explain conditional independence_ - Mathematics Stack Exchange.pdf │ └── P(B) probabilty and weighted avg of condtional probability.PNG └── Unit 4 Revise │ ├── Beaking sticks confusion.PNG │ ├── Bernoulli RV.PNG │ ├── Binomial RV.png │ ├── Conditional Expecation review all.png │ ├── Conditional Joint (Two Random Variable) PMF.PNG │ ├── Conditional PMF.PNG │ ├── Conditional more than two RV PMF.PNG │ ├── Conditional, PMF Expectation and Variance.PNG │ ├── Expecatation of the binomial.PNG │ ├── Expectation 1.png │ ├── Expectation Bernoulli.PNG │ ├── Expectation Linearity.PNG │ ├── Expectation Uniform.PNG │ ├── Expectation Weighted AVG.PNG │ ├── Expectation properties.PNG │ ├── Expected value of rule.PNG │ ├── Geometric PMF expecation memoryless and.PNG │ ├── Geometric RV PMF.PNG │ ├── Indepedent property.png │ ├── Independent best example.PNG │ ├── Joint PMF and Multiple RV with Specific PMF.png │ ├── Linearity Expectation │ ├── Lienearity of Expecatation 1.PNG │ └── Linearity of Expecatation 2.PNG │ ├── Multiple RV and Expecation Rule.PNG │ ├── PMF 1.PNG │ ├── PMF 2.PNG │ ├── PMF 3.PNG │ ├── Probablility cheatbook.png │ ├── Total epectation and two random variable of conditional expectation.PNG │ ├── Total expecation and probabilty reveiw all.PNG │ ├── Total expecation.PNG │ ├── Total expectation Example.png │ ├── Uniform RV.PNG │ ├── Variance Bernoulli.PNG │ ├── Variance of Bernoulli Random Variable with a Random Variable as parameter - Cross Validated.pdf │ ├── Variance of the uniform.PNG │ ├── Variance proof and linearity properties.PNG │ └── marginal of joint pmf.PNG ├── Books ├── Introduction to Probability, 2nd Edition.pdf └── Romeo and Juliet - Purdue Universtiy.pdf ├── Final Exam ├── 1 _ Final Exam _ 6.431x Courseware _ edX.pdf ├── 2 _ Final Exam _ 6.431x Courseware _ edX.pdf ├── 3 _ Final Exam _ 6.431x Courseware _ edX.pdf ├── 4 _ Final Exam _ 6.431x Courseware _ edX.pdf ├── 5 _ Final Exam _ 6.431x Courseware _ edX.pdf ├── 6 _ Final Exam _ 6.431x Courseware _ edX.pdf └── Exam Rules _ Final Exam _ 6.431x Courseware _ edX.pdf ├── MIT certificate.PNG ├── MidTerm Exam 1 ├── 1. _ Exam 1 _ 6.431x Courseware _ edX.pdf ├── 2. _ Exam 1 _ 6.431x Courseware _ edX.pdf ├── 3. _ Exam 1 _ 6.431x Courseware _ edX.pdf ├── 4. _ Exam 1 _ 6.431x Courseware _ edX.pdf ├── 5. _ Exam 1 _ 6.431x Courseware _ edX.pdf ├── 6. _ Exam 1 _ 6.431x Courseware _ edX.pdf └── Exam Rules _ Exam 1 _ 6.431x Courseware _ edX.pdf ├── MidTerm Exam 2 ├── 1 _ Exam 2 _ 6.431x Courseware _ edX.pdf ├── 2 _ Exam 2 _ 6.431x Courseware _ edX.pdf ├── 3 _ Exam 2 _ 6.431x Courseware _ edX.pdf ├── 4 _ Exam 2 _ 6.431x Courseware _ edX.pdf ├── 5 _ Exam 2 _ 6.431x Courseware _ edX.pdf └── Exam Rules _ Exam 2 _ 6.431x Courseware _ edX.pdf ├── README.md ├── Unit 1 Probability models and axioms ├── Lec. 1 Probability models and axioms │ ├── 10. Exercise_ Simple properties.pdf │ ├── 12. Exercise_ More properties.pdf │ ├── 14. Exercise_ Discrete probability calculations.pdf │ ├── 16. Exercise_ Continuous probability calculations.pdf │ ├── 18. Exercise_ Using countable additivity.pdf │ ├── 19. Exercise_ Uniform probabilities on the integers.pdf │ ├── 20. Exercise_ On countable additivity.pdf │ ├── 4. Exercise_ Sample space.pdf │ ├── 6. Exercise_ Tree representations.pdf │ └── 8. Exercise_ Axioms.pdf └── Problem Set 1 │ ├── 1. Venn diagrams.pdf │ ├── 2. Set operations and probabilities.pdf │ ├── 3. Three tosses of a fair coin.pdf │ ├── 4. Parking lot problem.pdf │ ├── 5. Probabilities on a continuous sample space.pdf │ └── 6. Upper and lower bounds on the probability of intersection.pdf ├── Unit 10 Markov chains ├── Lec. 24 Finite-state Markov chains │ ├── 12. Exercise_ Convergence.pdf │ ├── 14. Exercise_ Recurrent and transient states.pdf │ ├── 4. Exercise_ Checkout counter.pdf │ ├── 6. Exercise_ Markov property.pdf │ ├── 8. Exercise_ n-step recursion.pdf │ └── 9. Exercise_ n-step calculation.pdf ├── Lec. 25 Steady-state behavior of Markov chains │ ├── 10. Exercise_ Steady-state behavior.pdf │ ├── 12. Exercise_ Steady-state calculation.pdf │ ├── 14. Exercise_ Frequency interpretations.pdf │ ├── 16. Exercise_ Birth and death.pdf │ ├── 5. Exercise_ Path calculation.pdf │ └── 8. Exercise_ Periodic states.pdf ├── Lec. 26 Absorption probabilities and expected time to absorption │ ├── 11. Exercise_ Expected time to absorption.pdf │ ├── 13. Exercise_ Time until consecutive successes.pdf │ ├── 15. Exercise_ Gambler's ruin.pdf │ ├── 5. Exercise_ Steady-state approximation.pdf │ ├── 7. Exercise_ Design of a phone system.pdf │ └── 9. Exercise_ Probability of absorption.pdf └── Problem Set 10 │ ├── 1. Steady-state convergence.pdf │ ├── 2. Oscar's running shoes.pdf │ ├── 3. Checking the Markov property.pdf │ ├── 4. A simple Markov chain.pdf │ └── 5. Coin tosses revisited.pdf ├── Unit 2 Conditioning and independence ├── Lec. 2 Conditioning and Bayes' rule │ ├── 11. Exercise_ Total probability theorem.pdf │ ├── 13. Exercise_ Bayes' rule and the false-positive puzzle.pdf │ ├── 3. Exercise_ Conditional probabilities.pdf │ ├── 5. Exercise_ Conditional probabilities in a continuous model.pdf │ └── 9. Exercise_ The multiplication rule.pdf ├── Lec. 3 Independence │ ├── 10. Exercise_ Conditional independence.pdf │ ├── 13. Exercise_ Independence of multiple events.pdf │ ├── 16. Exercise_ Reliability.pdf │ ├── 4. Exercise_ Independence of two events - I.pdf │ ├── 5. Exercise_ Independence of two events - II.pdf │ ├── 6. Exercise_ Independence of two events - III.pdf │ └── 8. Exercise_ Independence of event complements.pdf └── Problem Set 2 │ ├── 1. Two five-sided dice.pdf │ ├── 2. A reliability problem.pdf │ ├── 3. Oscar's lost dog in the forest.pdf │ └── 4. Serap and her umbrella.pdf ├── Unit 3 Counting ├── Lec. 4 Counting │ ├── 10. Exercise_ Binomial probabilities.pdf │ ├── 12. Exercise_ Coin tossing.pdf │ ├── 14. Exercise_ Counting partitions.pdf │ ├── 4. Exercise_ Counting.pdf │ ├── 6. Exercise_ Use counting to calculate probabilities.pdf │ └── 8. Exercise_ Counting committees.pdf └── Problem Set 3 │ ├── 1. Customers arriving at a restaurant.pdf │ ├── 2. A three-sided die.pdf │ ├── 3. Forming a committee.pdf │ ├── 4. Proving binomial identities via counting.pdf │ └── 5. Hats in a box.pdf ├── Unit 4 Discrete random variables ├── Lec. 5 Probability mass functions and expectations │ ├── 11. Exercise_ The binomial PMF.pdf │ ├── 13. Exercise_ Geometric random variables.pdf │ ├── 15. Exercise_ Expectation calculation.pdf │ ├── 17. Exercise_ Random variables with bounded range.pdf │ ├── 19. Exercise_ The expected value rule.pdf │ ├── 21. Exercise_ Linearity of expectations.pdf │ ├── 3. Exercise_ Random variables.pdf │ ├── 5. Exercise_ PMF calculation.pdf │ ├── 6. Exercise_ Random variables versus numbers.pdf │ └── 8. Exercise_ Indicator variables.pdf ├── Lec. 6 Variance; Conditioning on an event; Multiple r.v.'s │ ├── 11. Exercise_ Total expectation calculation.pdf │ ├── 12. Exercise_ Memorylessness of the geometric.pdf │ ├── 14. Exercise_ Joint PMF calculation.pdf │ ├── 15. Exercise_ Expected value rule.pdf │ ├── 17. Exercise_ Linearity of expectations drill.pdf │ ├── 18. Exercise_ Using linearity of expectations.pdf │ ├── 3. Exercise_ Variance calculation.pdf │ ├── 4. Exercise_ Variance properties.pdf │ ├── 6. Exercise_ Variance of the uniform.pdf │ └── 8. Exercise_ Conditional variance.pdf ├── Lec. 7 Conditioning on a random variable; Independence of r.v.'s │ ├── 11. Exercise_ Independence and expectations.pdf │ ├── 13. Exercise_ Independence and variances.pdf │ ├── 15. Exercise_ The hat problem.pdf │ ├── 3. Exercise_ Conditional PMFs.pdf │ ├── 5. Exercise_ The expected value rule with conditioning.pdf │ ├── 7. Exercise_ Independence.pdf │ └── 8. Exercise_ A criterion for independence.pdf └── Problem Set 4 │ ├── 1. Tosses of a biased coin.pdf │ ├── 2. Three-sided dice.pdf │ ├── 3. PMF, expectation, and variance.pdf │ ├── 4. Joint PMF.pdf │ ├── 5. Indicator variables.pdf │ └── 6. True or False.pdf ├── Unit 5 Continuous random variables ├── Lec. 10 Conditioning on a random variable; Independence; Bayes' rule │ ├── 10. Exercise_ Independence and expectations II.pdf │ ├── 11. Exercise_ Independence and CDFs.pdf │ ├── 13. Exercise_ Stick-breaking.pdf │ ├── 15. Exercise_ Independent normals.pdf │ ├── 17. Exercise_ The discrete Bayes rule.pdf │ ├── 20. Exercise_ Discrete unknown, continuous measurement.pdf │ ├── 22. Exercise_ Inference of the bias of a coin.pdf │ ├── 3. Exercise_ Conditional PDF.pdf │ ├── 5. Exercise_ Conditional PDFs.pdf │ ├── 7. Exercise_ Expected value rule and total expectation.pdf │ └── 9. Exercise_ Definition of independence.pdf ├── Lec. 8 Probability density functions │ ├── 10. Exercise_ Exponential PDF.pdf │ ├── 12. Exercise_ Exponential CDF.pdf │ ├── 14. Exercise_ Normal random variables.pdf │ ├── 16. Exercise_ Using the normal tables.pdf │ ├── 3. Exercise_ PDFs.pdf │ ├── 5. Exercise_ Piecewise constant PDF.pdf │ └── 7. Exercise_ Uniform PDF.pdf ├── Lec. 9 Conditioning on an event; Multiple r.v.'s │ ├── 10. Exercise_ A mixed random variable.pdf │ ├── 12. Exercise_ Jointly continuous r.v.'s.pdf │ ├── 13. Exercise_ From joint PDFs to probabilities.pdf │ ├── 15. Exercise_ Finding a marginal PDF.pdf │ ├── 17. Exercise_ From joint PDFs to the marginals.pdf │ ├── 19. Exercise_ Joint CDFs.pdf │ ├── 3. Exercise_ A conditional PDF.pdf │ ├── 6. Exercise_ Memorylessness of the exponential.pdf │ └── 8. Exercise_ Total probability theorem II.pdf └── Problem Set 5 │ ├── 1. Normal random variables.pdf │ ├── 1. Standard normal table.pdf │ ├── 2. CDF.pdf │ ├── 3. A joint PDF given by a simple formula.pdf │ ├── 4. Sophia's vacation.pdf │ ├── 5. True or False.pdf │ ├── 6. Bayes' rule.pdf │ └── 7. A joint PDF on a triangular region.pdf ├── Unit 6 Further topics on random variables ├── Lec. 11 Derived distributions │ ├── 10. Exercise_ Using the formula for the monotonic case.pdf │ ├── 13. Exercise_ Nonmonotonic functions.pdf │ ├── 15. Exercise_ A function of multiple r.v.'s.pdf │ ├── 3. Exercise_ Linear functions of discrete r.v.'s.pdf │ ├── 5. Exercise_ Linear functions of continuous r.v.'s.pdf │ └── 8. Exercise_ PDF of a general function.pdf ├── Lec. 12 Sums of independent r.v.'s; Covariance and correlation │ ├── 11. Exercise_ Covariance properties.pdf │ ├── 13. Exercise_ The variance of a sum.pdf │ ├── 15. Exercise_ Correlation coefficient.pdf │ ├── 18. Exercise_ Correlation properties.pdf │ ├── 3. Exercise_ Discrete convolution.pdf │ ├── 5. Exercise_ Continuous convolution.pdf │ ├── 7. Exercise_ Sum of normals.pdf │ └── 9. Exercise_ Covariance calculation.pdf ├── Lec. 13 Conditional expectation and variance revisited │ ├── 10. Cond variance II.pdf │ ├── 11.Conditional varia.pdf │ ├── 15..pdf │ ├── 18.Second gen.pdf │ ├── 3. Condiotional Exp.pdf │ ├── 5. Iterated expe.pdf │ └── 7. Conditional expe.pdf └── Problem Set 6 │ ├── 1. The PDF of the logarithm of X.pdf │ ├── 2. Functions of the standard normal.pdf │ ├── 3. The PDF of the maximum.pdf │ ├── 4. Convolution calculations.pdf │ ├── 5. Covariance of the multinomial.pdf │ ├── 6. Correlation coefficients.pdf │ └── 7. Sum of a random number of r.v.'s.pdf ├── Unit 7 Bayesian inference ├── Lec. 14 Introduction to Bayesian inference │ ├── 10. Exercise_ Discrete unknown and continuous observation.pdf │ ├── 12. Exercise_ Continuous unknown and observation.pdf │ ├── 14. Exercise_ The posterior of a coin's bias.pdf │ ├── 16. Exercise_ Moments of the Beta distribution.pdf │ ├── 4. Exercise_ Hypothesis testing versus estimation.pdf │ ├── 6. Exercise_ Estimates and estimators.pdf │ └── 8. Exercise_ Discrete unknowns.pdf ├── Lec. 15 Linear models with normal noise │ ├── 10. Exercise_ The mean-squared error.pdf │ ├── 11. Exercise_ The effect of a stronger signal.pdf │ ├── 13. Exercise_ Multiple observations and unknowns.pdf │ ├── 3. Exercise_ Recognizing normal PDFs.pdf │ ├── 5. Exercise_ Normal unknown and additive noise.pdf │ ├── 7. Exercise_ Multiple observations.pdf │ └── 8. Exercise_ Multiple observations, more general model.pdf ├── Lec. 16 Least mean squares (LMS) estimation │ ├── 10. Exercise_ Mean squared error.pdf │ ├── 12. Exercise_ Multidimensional challenges.pdf │ ├── 14. Exercise_ Theoretical properties.pdf │ ├── 4. Exercise_ LMS estimation.pdf │ ├── 6. Exercise_ LMS estimation error.pdf │ └── 8. Exercise_ LMS example.pdf ├── Lec. 17 Linear least mean squares (LLMS) estimation │ ├── 1. Determining the type of a lightbulb.pdf │ ├── 11. Exercise_ Comparison for the coin problem.pdf │ ├── 13. Exercise_ LLMS with multiple observations.pdf │ ├── 16. Exercise_ Choice of representations.pdf │ ├── 3. Exercise_ LMS and LLMS.pdf │ ├── 5. Exercise_ LLMS without a constant term.pdf │ ├── 8. Exercise_ LLMS drill.pdf │ └── 9. Exercise_ Possible values of the estimates.pdf ├── Problem Set 7a │ ├── 1. Defective Coin.pdf │ ├── 2. Hypothesis test between two coins.pdf │ ├── 3. Hypothesis test with a continuous observation.pdf │ ├── 4. Trajectory estimation.pdf │ └── 5. Hypothesis test between two normals.pdf └── Problem Set 7b │ ├── 1. Determining the type of a lightbulb.pdf │ ├── 2. Estimating the parameter of a geometric r.v..pdf │ ├── 3. LLMS estimation.pdf │ ├── 4. LLMS estimation with random sums.pdf │ └── 5. Estimating the parameter of a uniform r.v..pdf ├── Unit 8 Limit theorems and classical statistics ├── Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers │ ├── 10. Exercise_ Polling.pdf │ ├── 13. Exercise_ Convergence in probability.pdf │ ├── 3. Exercise_ Markov inequality.pdf │ ├── 5. Exercise_ Chebyshev inequality.pdf │ ├── 6. Exercise_ Chebyshev versus Markov.pdf │ └── 8. Exercise_ Sample mean bounds.pdf ├── Lec. 19 The Central Limit Theorem (CLT) │ ├── 10. Exercise_ CLT for the binomial.pdf │ ├── 3. Exercise_ CLT.pdf │ ├── 5. Exercise_ CLT applicability.pdf │ └── 8. Exercise_ CLT practice.pdf ├── Lec. 20 An introduction to classical statistics │ ├── 1. Convergence in probability.pdf │ ├── 11. Exercise_ CI's via the CLT.pdf │ ├── 14. Exercise_ Natural estimators.pdf │ ├── 17. Exercise_ ML estimation.pdf │ ├── 4. Exercise_ Estimator properties.pdf │ ├── 7. Exercise_ Bias and MSE.pdf │ ├── 8. Exercise_ Confidence interval interpretation.pdf │ └── 9. Exercise_ A simple CI.pdf └── Problem Set 8 │ ├── 1. Convergence in probability.pdf │ ├── 2. Find the limits.pdf │ ├── 3. The sample mean.pdf │ ├── 4. Airline overbooking.pdf │ ├── 5. Maximum likelihood estimation.pdf │ └── 6. Tossing a triple of coins.pdf └── Unit 9 Bernoulli and Poisson processes ├── Lec. 21 The Bernoulli process ├── 11. Exercise_ Busy periods.pdf ├── 14. Exercise_ A variation on merging.pdf ├── 16. Exercise_ Splitting.pdf ├── 3. Exercise_ The Bernoulli process.pdf ├── 6. Exercise_ Time until the first failure.pdf ├── 8. Exercise_ Fresh start.pdf └── 9. Exercise_ More on fresh start.pdf ├── Lec. 22 The Poisson process ├── 12. Exercise_ Describing events.pdf ├── 13. Exercise_ Erlang r.v.'s.pdf ├── 14. Exercise_ The time of the kth arrival.pdf ├── 17. Exercise_ Bank tellers.pdf ├── 3. Exercise_ Poisson process definition.pdf ├── 5. Exercise_ Poisson models.pdf └── 9. Exercise_ Poisson practice.pdf ├── Lec. 23 More on the Poisson process ├── 10. Exercise_ Lightbulb burnouts.pdf ├── 11. Exercise_ The busy tellers.pdf ├── 13. Exercise_ At the coffee shop.pdf ├── 15. Exercise_ Random incidence.pdf ├── 17. Exercise_ Non-Poisson random incidence.pdf ├── 19. Exercise_ Sampling methods.pdf ├── 3. Exercise_ The sum of Poisson r.v.'s.pdf ├── 4. Exercise_ People in the park.pdf ├── 6. Exercise_ Processes in the park.pdf └── 8. Exercise_ What kind of people are they.pdf └── Problem Set 9 ├── 1. Marie gives away children toys.pdf ├── 2. Three engines.pdf ├── 3. Shuttles.pdf ├── 4. Ships.pdf ├── 5. Arrivals during overlapping time intervals.pdf ├── 6. Random incidence under Erlang interarrivals.pdf ├── 7. Sampling families.pdf └── 8. 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Link](https://credentials.edx.org/records/programs/shared/dd1f16009d8e4e54a762ce4a07020f45/) 5 | 6 | ![](MIT%20certificate.PNG) 7 | 8 | # 1. Introduction and Course Team 9 | 10 | ### Welcome to 6.431x, an introduction to probabilistic models, including random processes and the basic elements of statistical inference. 11 | 12 | The world is full of uncertainty: accidents, storms, unruly financial markets, noisy communications. The world is also full of data. Probabilistic modeling and the related field of statistical inference are the keys to analyzing data and making scientifically sound predictions. 13 | 14 | The course covers all of the basic probability concepts, including: 15 | 16 | * multiple discrete or continuous random variables, expectations, and conditional distributions 17 | 18 | * laws of large numbers 19 | 20 | * the main tools of Bayesian inference methods 21 | 22 | * an introduction to random processes (Poisson processes and Markov chains) 23 | 24 | ## 2. Course objectives 25 | Upon successful completion of this course, you will: 26 | 27 | ### At a conceptual level: 28 | 29 | * Master the basic concepts associated with probability models . 30 | 31 | * Be able to translate models described in words to mathematical ones. 32 | 33 | * Understand the main concepts and assumptions underlying Bayesian and classical inference . 34 | 35 | * Obtain some familiarity with the range of applications of inference methods . 36 | 37 | ### At a more technical level: 38 | 39 | * Become familiar with basic and common probability distributions . 40 | 41 | * Learn how to use conditioning to simplify the analysis of complicated models. 42 | 43 | * Have facility manipulating probability mass functions , densities , and expectations . 44 | 45 | * Develop a solid understanding of the concept of conditional expectation and its role in inference. 46 | 47 | * Understand the power of laws of large numbers and be able to use them when appropriate. 48 | 49 | * Become familiar with the basic inference methodologies (for both estimation and hypothesis testing ) and be able to apply them. 50 | 51 | * Acquire a good understanding of two basic stochastic processes (Bernoulli and Poisson) and their use in modeling. 52 | 53 | * Learn how to formulate simple dynamical models as Markov chains and analyze them. 54 | 55 | ## Professor John Tsitsiklis 56 | Dr. John Tsitsiklis is a Clarence J Lebel Professor in the Department of Electrical Engineering and Computer Science, and the director of the Laboratory for Information and Decision Systems at MIT. 57 | 58 | His research interests are in the fields of systems, optimization, control, and operations research. He is a coauthor of Parallel and Distributed Computation: Numerical Methods (1989, with D. Bertsekas), Neuro-Dynamic Programming (1996, with D. Bertsekas), Introduction to Linear Optimization (1997, with D. Bertsimas), and Introduction to Probability (1st ed. 2002, 2nd. ed. 2008, with D. Bertsekas). He is also a coinventor in seven awarded U.S. patents. 59 | 60 | He is a member of the National Academy of Engineering, and a Fellow of the IEEE (1999) and of INFORMS (2007). His distinctions include the ACM Sigmetrics Achievement Award (2016), the INFORMS John von Neumann Theory Prize (2018), and the IEEE Control Systems Award (2018). He holds honorary doctorates from the Universite catholique de Louvain, (2008), the Athens University of Economics and Business (2018), and the Harokopio University. 61 | 62 | Professor Tsitsiklis has been teaching probability for over 20 years. 63 | 64 | 65 | ## Professor Patrick Jaillet 66 | Patrick Jaillet is Dugald C. Jackson Professor in the Department of Electrical Engineering and Computer Science and a member of the Laboratory for Information and Decision Systems at MIT. 67 | 68 | Professor Jaillet's research interests include online optimization and learning; machine learning; and decision making under uncertainty. Professor Jaillet's teaching covers subjects such as machine learning; algorithms; mathematical programming; network science and models; and probability. Dr. Jaillet's consulting activities primarily focus on the development of optimization-based analytic solutions in various industries, including defense, financial, electronic marketplace, and information technology. 69 | 70 | Professor Jaillet was a fulbright scholar in 1990 and the recipient of many research and teaching awards. He is a Fellow of the Institute for Operations Research and Management Science Society (INFORMS), a member of the Mathematical Optimization Society (MOS), and a member of the Society for Industrial and Applied Mathematics (SIAM). He is currently an Associate Editor for INFORMS Journal on Optimization, Networks, and Naval Research Logistics, and has been an Associate Editor for Operations Research from 1994 until 2005 and for Transportation Science from 2002 until 2017. 71 | 72 | ## Professor Dimitri Bertsekas 73 | Dimitri P. Bertsekas is McAfee Professor of Engineering in the Electrical Engineering and Computer Science Department of MIT. In 2019, he was also appointed a full time professor in the department of Computer, Information, and Decision Systems Engineering at Arizona State University, Tempe, while maintaining a research position at MIT. 74 | 75 | His research spans several fields, including optimization, control, large-scale computation, and data communication networks, and is closely tied to his teaching and book authoring activities. He has written numerous research papers, and seventeen books and research monographs, several of which are used as textbooks in MIT classes. 76 | 77 | Professor Bertsekas was awarded the INFORMS 1997 Prize for Research Excellence in the Interface Between Operations Research and Computer Science for his book "Neuro-Dynamic Programming", the 2000 Greek National Award for Operations Research, the 2001 ACC John R. Ragazzini Education Award, the 2009 INFORMS Expository Writing Award, the 2014 ACC Richard E. Bellman Control Heritage Award for "contributions to the foundations of deterministic and stochastic optimization-based methods in systems and control," the 2014 Khachiyan Prize for Life-Time Accomplishments in Optimization, and the SIAM/MOS 2015 George B. Dantzig Prize. In 2018, he was awarded, jointly with his coauthor John Tsitsiklis, the INFORMS John von Neumann Theory Prize, for the contributions of the research monographs "Parallel and Distributed Computation" and "Neuro-Dynamic Programming". In 2001, he was elected to the United States National Academy of Engineering for "pioneering contributions to fundamental research, practice and education of optimization/control theory, and especially its application to data communication networks." 78 | 79 | Prof Bertsekas has been teaching probability for over 15 years. 80 | 81 | ## 3. Study guide 82 | 83 | A guide on how to use the wealth of available material 84 | 85 | This class provides you with a great wealth of material, perhaps more than you can fully digest. This “guide" offers some tips about how to use this material. 86 | 87 | Start with the overview of a unit, when available. This will help you get an overview of what is to happen next. Similarly, at the end of a unit, watch the unit summary to consolidate your understanding of the “big picture" and of the relation between different concepts. 88 | 89 | Watch the lecture videos. You may want to download the slides (clean or annotated) at the beginning of each lecture, especially if you cannot receive high-quality streaming video. Some of the lecture clips proceed at a moderate speed. Whenever you feel comfortable, you may want to speed up the video and run it faster, at 1.5x. 90 | 91 | Do the exercises! The exercises that follow most of the lecture clips are a most critical part of this class. Some of the exercises are simple adaptations of you may have just heard. Other exercises will require more thought. Do your best to solve them right after each clip — do not defer this for later – so that you can consolidate your understanding. After your attempt, whether successful or not, do look at the solutions, which you will be able to see as soon as you submit your own answers. 92 | 93 | Solved problems and additional materials. In most of the units, we are providing you with many problems that are solved by members of our staff. We provide both video clips and written solutions. Depending on your learning style, you may pick and choose which format to focus on. But in either case, it is important that you get exposed to a large number of problems. 94 | 95 | The textbook. If you have access to the textbook, you can find more precise statements of what was discussed in lecture, additional facts, as well as several examples. While the textbook is recommended, the materials provided by this course are self-contained. See the “Textbook information" tab in Unit 0 for more details. 96 | 97 | Problem sets. One can really master the subject only by solving problems – a large number of them. Some of the problems will be straightforward applications of what you have learned. A few of them will be more challenging. Do not despair if you cannot solve a problem – no one is expected to do everything perfectly. However, once the problem set solutions are released (which will happen on the due date of the problem set), make sure to go over the solutions to those problems that you could not solve correctly. 98 | 99 | Exams. The midterm exams are designed so that in an on-campus version, learners would be given two hours. The final exam is designed so that in an on-campus version, learners would be given three hours. You should not expect to spend much more than this amount of time on them. In this respect, those weeks that have exams (and no problem sets!) will not have higher demands on your time. The level of difficulty of exam questions will be somewhere between the lecture exercises and homework problems. 100 | 101 | Time management. The corresponding on-campus class is designed so that students with appropriate prerequisites spend about 12 hours each week on lectures, recitations, readings, and homework. You should expect a comparable effort, or more if you need to catch up on background material. In a typical week, there will be 2 hours of lecture clips, but it might take you 4-5 hours when you add the time spent on exercises. Plan to spend another 3-4 hours watching solved problems and additional materials, and on textbook readings. Finally, expect about 4 hours spent on the weekly problem sets. 102 | 103 | Additional practice problems. For those of you who wish to dive even deeper into the subject, you can find a good collection of problems at the end of each chapter of the print edition of the book, whose solutions are available online. 104 | -------------------------------------------------------------------------------- /Unit 1 Probability models and axioms/Lec. 1 Probability models and axioms/10. Exercise_ Simple properties.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 1 Probability models and axioms/Lec. 1 Probability models and axioms/10. Exercise_ Simple properties.pdf -------------------------------------------------------------------------------- /Unit 1 Probability models and axioms/Lec. 1 Probability models and axioms/12. 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Exercise_ The multiplication rule.pdf -------------------------------------------------------------------------------- /Unit 2 Conditioning and independence/Lec. 3 Independence/10. Exercise_ Conditional independence.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 2 Conditioning and independence/Lec. 3 Independence/10. Exercise_ Conditional independence.pdf -------------------------------------------------------------------------------- /Unit 2 Conditioning and independence/Lec. 3 Independence/13. Exercise_ Independence of multiple events.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 2 Conditioning and independence/Lec. 3 Independence/13. 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Exercise_ Independence and variances.pdf -------------------------------------------------------------------------------- /Unit 4 Discrete random variables/Lec. 7 Conditioning on a random variable; Independence of r.v.'s/15. Exercise_ The hat problem.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 4 Discrete random variables/Lec. 7 Conditioning on a random variable; Independence of r.v.'s/15. Exercise_ The hat problem.pdf -------------------------------------------------------------------------------- /Unit 4 Discrete random variables/Lec. 7 Conditioning on a random variable; Independence of r.v.'s/3. 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Joint PMF.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 4 Discrete random variables/Problem Set 4/4. Joint PMF.pdf -------------------------------------------------------------------------------- /Unit 4 Discrete random variables/Problem Set 4/5. Indicator variables.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 4 Discrete random variables/Problem Set 4/5. Indicator variables.pdf -------------------------------------------------------------------------------- /Unit 4 Discrete random variables/Problem Set 4/6. True or False.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 4 Discrete random variables/Problem Set 4/6. True or False.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Lec. 10 Conditioning on a random variable; Independence; Bayes' rule/10. Exercise_ Independence and expectations II.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Lec. 10 Conditioning on a random variable; Independence; Bayes' rule/10. 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Exercise_ Exponential PDF.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Lec. 8 Probability density functions/10. Exercise_ Exponential PDF.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Lec. 8 Probability density functions/12. Exercise_ Exponential CDF.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Lec. 8 Probability density functions/12. Exercise_ Exponential CDF.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Lec. 8 Probability density functions/14. 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Exercise_ Uniform PDF.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Lec. 8 Probability density functions/7. Exercise_ Uniform PDF.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Lec. 9 Conditioning on an event; Multiple r.v.'s/10. Exercise_ A mixed random variable.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Lec. 9 Conditioning on an event; Multiple r.v.'s/10. Exercise_ A mixed random variable.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Lec. 9 Conditioning on an event; Multiple r.v.'s/12. 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Sophia's vacation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Problem Set 5/4. Sophia's vacation.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Problem Set 5/5. True or False.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Problem Set 5/5. True or False.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Problem Set 5/6. Bayes' rule.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Problem Set 5/6. Bayes' rule.pdf -------------------------------------------------------------------------------- /Unit 5 Continuous random variables/Problem Set 5/7. A joint PDF on a triangular region.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 5 Continuous random variables/Problem Set 5/7. A joint PDF on a triangular region.pdf -------------------------------------------------------------------------------- /Unit 6 Further topics on random variables/Lec. 11 Derived distributions/10. 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Iterated expe.pdf -------------------------------------------------------------------------------- /Unit 6 Further topics on random variables/Lec. 13 Conditional expectation and variance revisited/7. Conditional expe.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 6 Further topics on random variables/Lec. 13 Conditional expectation and variance revisited/7. Conditional expe.pdf -------------------------------------------------------------------------------- /Unit 6 Further topics on random variables/Problem Set 6/1. The PDF of the logarithm of X.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 6 Further topics on random variables/Problem Set 6/1. 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Exercise_ Multiple observations.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 15 Linear models with normal noise/7. Exercise_ Multiple observations.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 15 Linear models with normal noise/8. Exercise_ Multiple observations, more general model.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 15 Linear models with normal noise/8. Exercise_ Multiple observations, more general model.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/10. Exercise_ Mean squared error.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/10. Exercise_ Mean squared error.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/12. Exercise_ Multidimensional challenges.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/12. Exercise_ Multidimensional challenges.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/14. Exercise_ Theoretical properties.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/14. Exercise_ Theoretical properties.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/4. Exercise_ LMS estimation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/4. Exercise_ LMS estimation.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/6. Exercise_ LMS estimation error.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/6. Exercise_ LMS estimation error.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/8. Exercise_ LMS example.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 16 Least mean squares (LMS) estimation/8. Exercise_ LMS example.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/1. Determining the type of a lightbulb.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/1. Determining the type of a lightbulb.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/11. Exercise_ Comparison for the coin problem.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/11. Exercise_ Comparison for the coin problem.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/13. Exercise_ LLMS with multiple observations.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/13. Exercise_ LLMS with multiple observations.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/16. Exercise_ Choice of representations.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/16. Exercise_ Choice of representations.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/3. Exercise_ LMS and LLMS.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/3. Exercise_ LMS and LLMS.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/5. Exercise_ LLMS without a constant term.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/5. Exercise_ LLMS without a constant term.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/8. Exercise_ LLMS drill.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/8. Exercise_ LLMS drill.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/9. Exercise_ Possible values of the estimates.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Lec. 17 Linear least mean squares (LLMS) estimation/9. Exercise_ Possible values of the estimates.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7a/1. Defective Coin.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7a/1. Defective Coin.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7a/2. Hypothesis test between two coins.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7a/2. Hypothesis test between two coins.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7a/3. Hypothesis test with a continuous observation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7a/3. Hypothesis test with a continuous observation.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7a/4. Trajectory estimation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7a/4. Trajectory estimation.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7a/5. Hypothesis test between two normals.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7a/5. Hypothesis test between two normals.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7b/1. Determining the type of a lightbulb.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7b/1. Determining the type of a lightbulb.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7b/2. Estimating the parameter of a geometric r.v..pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7b/2. Estimating the parameter of a geometric r.v..pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7b/3. LLMS estimation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7b/3. LLMS estimation.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7b/4. LLMS estimation with random sums.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7b/4. LLMS estimation with random sums.pdf -------------------------------------------------------------------------------- /Unit 7 Bayesian inference/Problem Set 7b/5. Estimating the parameter of a uniform r.v..pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 7 Bayesian inference/Problem Set 7b/5. Estimating the parameter of a uniform r.v..pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/10. Exercise_ Polling.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/10. Exercise_ Polling.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/13. Exercise_ Convergence in probability.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/13. Exercise_ Convergence in probability.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/3. Exercise_ Markov inequality.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/3. Exercise_ Markov inequality.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/5. Exercise_ Chebyshev inequality.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/5. Exercise_ Chebyshev inequality.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/6. Exercise_ Chebyshev versus Markov.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/6. Exercise_ Chebyshev versus Markov.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/8. Exercise_ Sample mean bounds.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 18 Inequalities, convergence, and the Weak Law of Large Numbers/8. Exercise_ Sample mean bounds.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/10. Exercise_ CLT for the binomial.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/10. Exercise_ CLT for the binomial.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/3. Exercise_ CLT.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/3. Exercise_ CLT.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/5. Exercise_ CLT applicability.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/5. Exercise_ CLT applicability.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/8. Exercise_ CLT practice.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 19 The Central Limit Theorem (CLT)/8. Exercise_ CLT practice.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/1. Convergence in probability.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/1. Convergence in probability.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/11. Exercise_ CI's via the CLT.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/11. Exercise_ CI's via the CLT.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/14. Exercise_ Natural estimators.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/14. Exercise_ Natural estimators.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/17. Exercise_ ML estimation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/17. Exercise_ ML estimation.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/4. Exercise_ Estimator properties.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/4. Exercise_ Estimator properties.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/7. Exercise_ Bias and MSE.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/7. Exercise_ Bias and MSE.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/8. Exercise_ Confidence interval interpretation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/8. Exercise_ Confidence interval interpretation.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/9. Exercise_ A simple CI.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Lec. 20 An introduction to classical statistics/9. Exercise_ A simple CI.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Problem Set 8/1. Convergence in probability.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Problem Set 8/1. Convergence in probability.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Problem Set 8/2. Find the limits.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Problem Set 8/2. Find the limits.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Problem Set 8/3. The sample mean.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Problem Set 8/3. The sample mean.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Problem Set 8/4. Airline overbooking.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Problem Set 8/4. Airline overbooking.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Problem Set 8/5. Maximum likelihood estimation.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Problem Set 8/5. Maximum likelihood estimation.pdf -------------------------------------------------------------------------------- /Unit 8 Limit theorems and classical statistics/Problem Set 8/6. Tossing a triple of coins.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 8 Limit theorems and classical statistics/Problem Set 8/6. Tossing a triple of coins.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/11. Exercise_ Busy periods.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/11. Exercise_ Busy periods.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/14. Exercise_ A variation on merging.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/14. Exercise_ A variation on merging.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/16. Exercise_ Splitting.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/16. Exercise_ Splitting.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/3. Exercise_ The Bernoulli process.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/3. Exercise_ The Bernoulli process.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/6. Exercise_ Time until the first failure.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/6. Exercise_ Time until the first failure.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/8. Exercise_ Fresh start.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/8. Exercise_ Fresh start.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/9. Exercise_ More on fresh start.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 21 The Bernoulli process/9. Exercise_ More on fresh start.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/12. Exercise_ Describing events.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/12. Exercise_ Describing events.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/13. Exercise_ Erlang r.v.'s.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/13. Exercise_ Erlang r.v.'s.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/14. Exercise_ The time of the kth arrival.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/14. Exercise_ The time of the kth arrival.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/17. Exercise_ Bank tellers.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/17. Exercise_ Bank tellers.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/3. Exercise_ Poisson process definition.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/3. Exercise_ Poisson process definition.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/5. Exercise_ Poisson models.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/5. Exercise_ Poisson models.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/9. Exercise_ Poisson practice.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 22 The Poisson process/9. Exercise_ Poisson practice.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/10. Exercise_ Lightbulb burnouts.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/10. Exercise_ Lightbulb burnouts.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/11. Exercise_ The busy tellers.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/11. Exercise_ The busy tellers.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/13. Exercise_ At the coffee shop.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/13. Exercise_ At the coffee shop.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/15. Exercise_ Random incidence.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/15. Exercise_ Random incidence.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/17. Exercise_ Non-Poisson random incidence.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/17. Exercise_ Non-Poisson random incidence.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/19. Exercise_ Sampling methods.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/19. Exercise_ Sampling methods.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/3. Exercise_ The sum of Poisson r.v.'s.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/3. Exercise_ The sum of Poisson r.v.'s.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/4. Exercise_ People in the park.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/4. Exercise_ People in the park.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/6. Exercise_ Processes in the park.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/6. Exercise_ Processes in the park.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/8. Exercise_ What kind of people are they.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Lec. 23 More on the Poisson process/8. Exercise_ What kind of people are they.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/1. Marie gives away children toys.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/1. Marie gives away children toys.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/2. Three engines.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/2. Three engines.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/3. Shuttles.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/3. Shuttles.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/4. Ships.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/4. Ships.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/5. Arrivals during overlapping time intervals.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/5. Arrivals during overlapping time intervals.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/6. Random incidence under Erlang interarrivals.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/6. Random incidence under Erlang interarrivals.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/7. Sampling families.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/7. Sampling families.pdf -------------------------------------------------------------------------------- /Unit 9 Bernoulli and Poisson processes/Problem Set 9/8. Poisson fun.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/IggyZhao/Prob-Class-Notes/978d2f35c8c971d56bda93a41276afbf94beed6a/Unit 9 Bernoulli and Poisson processes/Problem Set 9/8. Poisson fun.pdf --------------------------------------------------------------------------------