# MM311: Mathematics 4B

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MM311: Mathematics 4B

## 1. 0. Probability and Statistics

### 1.1. 53.202 Materials

1.1.1. Lecture Notes

1.1.2. Worked Examples

1.1.3. Tutorials and Solutions

1.1.4. Past Papers

### 1.2. Presentation and Summarisation of Data

1.2.1. Hisograms

1.2.2. Stem and Leaf Plots

1.2.3. Box Plots

1.2.5. Introduction to Exploratory Data

1.2.6. Lecture Slides

1.2.6.1. Lecture 1

1.2.6.2. Lecture 2

### 1.3. RandomVariables

1.3.1. General Results

1.3.2. Discrete and Continuous Random Variables

1.3.3. Expectation

1.3.4. Variance

1.3.5. Moments

1.3.6. Quartiles

1.3.7. Mean and Variance of Linear Combinaions

1.3.8. Lecture Slides

1.3.8.1. Lecture 5

1.3.8.2. Lecture 6

### 1.4. Basic Laws

1.4.1. P(A or B)

1.4.2. P(not A)

1.4.3. P(A and B)

1.4.4. P(A|B)

1.4.5. Independence

1.4.6. Bayes Theorem

1.4.7. Lecture Slides

1.4.7.1. Lecture 4

1.4.7.2. Lecture 5

### 1.5. Probability Theory

1.5.1. Introduction

1.5.2. Origin of Probability

1.5.3. Assignment of Probability

1.5.4. Disributions

1.5.4.1. Poisson

1.5.4.2. Binomial

1.5.4.3. Geometric

1.5.4.4. Negative ~Exponential

1.5.4.5. Nomal

1.5.5. Lecture Slides

1.5.5.1. Lecture 3

1.5.5.2. Lecture 7

1.5.5.3. Lecture 8

1.5.5.4. Lecture 9

1.6.1. Variance

1.6.2. Standard Deviation

1.6.3. Range

1.6.4. Quartiles

1.6.5. Semi-interquartile Range

1.6.6. Lecture Slides

1.6.6.1. Lecture 6

1.7.1. Mean

1.7.2. Median

1.7.3. Mode

### 1.8. Z-test, samples and confidence

1.8.1. Lecture Slides

1.8.1.1. Lecture 10

1.8.1.2. Lecture 11

1.8.1.3. Lecture 12

## 2. 1. Fourier Series

### 2.5. Half Range Fourier Series

2.5.1. Fourier Sine Series

2.5.2. Fourier Cosine Series

2.9.1. Solutions

## 3. 4. Techniques of Complex Integration

3.1.1. Poles

### 3.2. Residue Integration

3.2.1. Method

3.2.2. Theorem

3.2.3. Residue at Simple Pole

3.2.3.1. 1st Method

3.2.3.2. 2nd Method

3.2.4. Residue at nth Order Pole

### 3.3. Real Integration using Residue Theorem

3.3.1. Integrals of Rational Functions

3.3.2. Improper Integrals

3.3.3. Fourier Integrals

3.4.1. Solutions

## 4. 5. Fourier Transforms

### 4.3. Properties of the Fourier Transform

4.3.1. Linearity

4.3.2. Derivative

4.3.3. Time-Shift

4.3.4. Frequency-Shift

4.3.5. Duality

4.3.6. Convolution of Two Functions

4.3.6.1. In Time

4.3.6.2. In Frequency

4.3.7. Multiplication

4.8.1. Solutions

5.1.1. Solutions

5.2.1. Solutions

5.3.1. Solutions

## 6. 2. Partial Differential Equations

### 6.3. Real Examples of PDE's

6.3.1. Heat Equation

6.3.2. Wave Equation

6.7.1. Solutions

## 7. 3. Complex Analysis

### 7.8. Functions of Complex Variables

7.8.1. Complex Exponential

7.8.2. Complex Trig & Hyperbolic Functions

7.8.3. Complex Logarithm

### 7.12. Cauchy's Integral Formula

7.12.1. CIF for Derivatives of Analytic Functions

### 7.13. Tutorials

7.13.1. Solutions (Part 1)

7.13.2. Solutions (Part 2)

8.1.1. Solutions