6.041SC | Fall 2013 | Undergraduate

# Probabilistic Systems Analysis and Applied Probability

## Unit IV: Laws Of Large Numbers And Inference

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In this section, we start with a discussion of limit theorems: the weak law of large numbers and the central limit theorem. We then introduce the subject of inference (estimation and hypothesis testing), from two alternative viewpoints: first, Bayesian inference, which relies on a prior distribution for unknown quantities and on the Bayes rule to incorporate new evidence; and, second, classical inference, in which no probabilistic assumptions are made on the unknown quantities and instead relies heavily on the laws of large numbers to provide statistical guarantees, e.g., in the form of confidence intervals.

Looking for something specific in this course? The Resource Index compiles links to most course resources in a single page.

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### Lecture Topics

 The sample mean A tool: Chebyshev’s inequality Convergence “in probability” Convergence of the sample mean (weak law of large numbers)

### Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. Two of the problems have an accompanying video where a teaching assistant solves the same problem.

#### Recitation Help Videos

PROBLEM # PROBLEM TITLE PROBLEM SOLVED BY
2

Convergence in Probability and in the Mean Part 1

Convergence in Probability and in the Mean Part 2

Kuang Xu

Kuang Xu

3 Convergence in Probability Example Kuang Xu

### Tutorial Problems

Review the tutorial problems in the PDF file below and try to solve them on your own.

### Problem Set and Solutions

Work the problems on your own and check your answers when you’re done. Problem set 9 covers Lectures 18 and 19.

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### Lecture Topics

 The Central Limit Theorem (CLT) CLT interpretation Application to polling CLT refinements

### Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. Two of the problems have an accompanying video where a teaching assistant solves the same problem.

#### Recitation Help Videos

PROBLEM # PROBLEM TITLE PROBLEM SOLVED BY
1 Probabilty Bounds Kuang Xu
2 Using the Central Limit Theorem Jagdish Ramakrishnan

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### Lecture Topics

 Inference applications Classical versus Bayesian satistics Maximum a posteriori probability estimation (Bayesian) Least mean squares estimation

### Lecture Activities

Note: There are no recitations or tutorials associated with this lecture.

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### Lecture Topics

 More on least means squares (LMS) estimation An example The mean squared error Theoretical properties (Bayesian) Linear LMS estimation

### Recitation Problems and Recitation Help Videos

Review the recitation problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem.

#### Recitation Help Videos

PROBLEM # PROBLEM TITLE PROBLEM SOLVED BY
1

Inferring a Parameter of Uniform Part 1

Inferring a Parameter of Uniform Part 2

Jimmy Li

Jimmy Li

### Tutorial Problems and Tutorial Help Videos

Review the tutorial problems in the PDF file below and try to solve them on your own. One of the problems has an accompanying video where a teaching assistant solves the same problem.

#### Tutorial Help Videos

PROBLEM # PROBLEM TITLE PROBLEM SOLVED BY
1 An Inference Example Jimmy Li

### Problem Set and Solutions

Work the problems on your own and check your answers when you’re done. Problem set 10 covers Lectures 20, 21, and 22.

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### Lecture Topics

 Classical statistics Maximum likelihood (ML) estimation Estimating a sample mean Confidence intervals (CIs) CIs using an estimated variance

### Recitation Problems

Review the recitation problems in the PDF file below and try to solve them on your own.

### Tutorial Problems

Review the tutorial problems in the PDF file below and try to solve them on your own.

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### Lecture Topics

 Review Maximum likelihood estimation Confidence intervals Linear regression Binary hypothesis testing Types of error Likelihood ratio test (LRT)

### Recitation Problems

Review the recitation problems in the PDF file below and try to solve them on your own.

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 Review of simple binary hypothesis tests Examples Testing composite hypotheses Is my coin fair? Is my die fair? Goodness of fit tests

### Problem Set and Solutions

Work the problems on your own and check your answers when you’re done. Problem set 11 covers Lectures 23, 24, and 25.

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## Course Info

Fall 2013
##### Learning Resource Types
Lecture Videos
Recitation Videos
Problem Sets with Solutions
Exams with Solutions