1.1. Sample spaces and events
1.2. Conditional probability and Independence
1.3. Bayes’ rule
1.4. The Inclusion-Exclusion Formula
1.5. Expectation
1.6. Variance
1.7. Chebyshev’s inequality
1.8. Weak law of large numbers
- Pigeonhole Principle (PDF)
- Probabilistic Method - Van der Waerden’s Theorem (PDF)
- Chernoff Bounds (PDF)
4.1. Preliminaries
4.2. Deviation of a sum on independent random variables
4.3. Chernoff Bound
4.4. Other versions of Chernoff Bound
5.1. A game
5.2. Introduction
5.3. Seeking the very best out of n possibilities
5.4. Calculating probabilities for choosing the very best on a spread-sheet
5.5. Seeking one of the top two (or top L) choices
5.6. Seeking the Best Expected Rank
6.1. Some Catalan families
6.2. Coding
6.3. Random sampling
7.1. What is a generating function?
7.2. Operations on classes and generating functions
7.3. Number of ways of giving change
7.4. Dots and Dashes
7.5. Generalized Recurrence Equations
7.6. Chord diagrams
7.7. Diagonals in Pascal’s triangle
7.8. Exponential generating functions
8.1. Basics
8.2. The Simplex Method
8.3. Linear Programming in Matrix Form
8.4. Duality
8.5. Zero-Sum Matrix Games
9.1. The maximum flow problem
9.2. Expressing as a linear program
9.3. Augmenting path algorithm
9.4. Dual
10.1. The task of sorting
10.2. Some sorting algorithms
11.1. The problem of finding the median
11.2. A good strategy?
11.3. Showing the Procedure is Linear
11.4. Improving the procedure
12.1. Randomized Median Finding
12.2. QUICKSORT
13.1. Batcher’s Algorithm
14.1. Euclid’s Algorithm
14.2. Modular Arithmetic
14.3. Groups
14.4. Subgroups and cosets
14.5. The Chinese Remainder Theorem
14.6. Fields and algebraic equations
15.1. Public Key Cryptosystems
15.2. The RSA code
15.3. Raising a Number to a High Power
15.4. Primality Testing
17.1. Motivation: Fourier Series
17.2. The Discrete Fourier Transform
17.3. Computing the discrete Fourier transform
17.4. Computing convolutions of sequences using Fast Fourier Transform
17.5. Fourier transforms modulo p and fast integer multiplication
18.1. Some History
18.2. Random data and compression
18.3. Shannon’s entropy Theorem
20.1. The LZ78 Algorithm
20.2. Worst-Case Analysis
20.3. Average-Case Analysis
21.1. Channel Coding
21.2. Binary symmetric channels
22.1. Linear Error Correcting Codes
22.2. Hamming Code
22.3. Perfect Codes
