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# MM311: Mathematics 4B

## 0. Probability and Statistics

### 53.202 Materials

Lecture Notes

Worked Examples

Tutorials and Solutions

Past Papers

### Presentation and Summarisation of Data

Hisograms

Stem and Leaf Plots

Box Plots

Introduction to Exploratory Data

Lecture Slides, Lecture 1, Lecture 2

### RandomVariables

General Results

Discrete and Continuous Random Variables

Expectation

Variance

Moments

Quartiles

Mean and Variance of Linear Combinaions

Lecture Slides, Lecture 5, Lecture 6

### Basic Laws

P(A or B)

P(not A)

P(A and B)

P(A|B)

Independence

Bayes Theorem

Lecture Slides, Lecture 4, Lecture 5

### Probability Theory

Introduction

Origin of Probability

Assignment of Probability

Disributions, Poisson, Binomial, Geometric, Negative ~Exponential, Nomal

Lecture Slides, Lecture 3, Lecture 7, Lecture 8, Lecture 9

Variance

Standard Deviation

Range

Quartiles

Semi-interquartile Range

Lecture Slides, Lecture 6

Mean

Median

Mode

### Z-test, samples and confidence

Lecture Slides, Lecture 10, Lecture 11, Lecture 12

Solutions

Solutions

Solutions

## 1. Fourier Series

### Half Range Fourier Series

Fourier Sine Series

Fourier Cosine Series

Solutions

Heat Equation

Wave Equation

Solutions

## 3. Complex Analysis

### Functions of Complex Variables

Complex Exponential

Complex Trig & Hyperbolic Functions

Complex Logarithm

### Cauchy's Integral Formula

CIF for Derivatives of Analytic Functions

### Tutorials

Solutions (Part 1)

Solutions (Part 2)

## 4. Techniques of Complex Integration

Poles

### Residue Integration

Method

Theorem

Residue at Simple Pole, 1st Method, 2nd Method

Residue at nth Order Pole

### Real Integration using Residue Theorem

Integrals of Rational Functions

Improper Integrals

Fourier Integrals

Solutions

## 5. Fourier Transforms

### Properties of the Fourier Transform

Linearity

Derivative

Time-Shift

Frequency-Shift

Duality

Convolution of Two Functions, In Time, In Frequency

Multiplication

Solutions

Solutions