Contents (38 pages)
VARIABLES- QUALITATIVE AND QUANTITATIVE
  1.1 Qualitative Data (Categorical Variables or Attributes)
  1.2 Quantitative Data
DESCRIPTIVE STATISTICS
  2.1 Sample Data versus Population Data
  2.2 Parameters and Statistics
  2.3 Location Statistics (measures of central tendency)
  2.4 Dispersion Statistics (measures of variability)
FREQUENCY DISTRIBUTIONS
  3.1 Frequency Measures
  3.2 Histogram
  3.3 Discrete Histogram
  3.4 Continuous Data Histogram
PROBABILITY AND ERROR
  4.1 Probability Properties
  4.2 Type I and II Errors.
  4.3 p-values and statistical significance
NORMAL DISTRIBUTION
  5.1 Properties of the Normal Distribution
  5.2 Estimating Probabilities Using Normal Distribution
  5.3 Calculating Parts Per Million Defects Given Normal Distribution
LINEAR REGRESSION ANALYSIS
  6.1 General Regression equation
  6.2 Simple linear regression
  6.3 Correlation
  6.4 Using Scatter Plots to Show Linear Relationships
  6.5 Multiple linear regression
Appendices:
A – Practice Test
B – Normal Distribution Tables
C – Useful Excel Functions

Contents: (47 pages)
1 Preliminaries
   1.1 Relations and Functions
   1.2 Counting and comparing infinite sets
   1.3 Principle of Induction
   1.3.1 Two kinds of induction proofs
2 Basic Counting
   2.1 Permutation and Combinations
   2.2 Distribution problems
3 Introduction to Graphs
   3.1 Representation of graphs
   3.2 Reachability in graphs
      3.2.1 Tours and cycles
      3.2.2 Connectivity
      3.2.3 k-connectivity
   3.3 Some special classes of graphs
   3.4 Problem Set
4 Counting techniques
   4.1 The pigeon hole principle
   4.2 Principle of Inclusion and Exclusion
   4.3 The probabilistic method
   4.4 Problem Set
   4.5 Some basics of probability theory
5 Recurrences and generating functions
   5.1 An iterative method - summation
   5.2 Linear recurrence equations
      5.2.1 Homogeneous equations
      5.2.2 Inhomogeneous equations
   5.3 Generating functions
      5.3.1 Binomial theorem
   5.4 Exponential generating functions
   5.5 Recurrences with two variables
   5.6 Probability generating functions
      5.6.1 Probabilistic inequalities
   5.7 Problem Set
6 Modular Arithmetic
   6.1 Divisibility
   6.2 Congruences
   6.3 Problem Set

Contents: (520 pages)
1 Discrete Probability Distributions
2 Continuous Probability Densities
3 Combinatorics
4 Conditional Probability
5 Distributions and Densities
6 Expected Value and Variance
7 Sums of Random Variables
8 Law of Large Numbers
9 Central Limit Theorem
10 Generating Functions
11 Markov Chains
12 Random Walks

Contents: (302 pages)
1 Modelling Uncertainty
2 Probability Space
3 Conditional Probability and Independence
4 Random Variable
5 Random Variables
6 Conditional Expectation
7 Gaussian Random Variables
8 Detection and Hypothesis Testing
9 Estimation
10 Limits of Random Variables
11 Law of Large Numbers & Central Limit Theorem
12 Random Processes Bernoulli - Poisson
13 Filtering Noise
14 Markov Chains - Discrete Time
15 Markov Chains - Continuous Time
16 Applications

Contents: (49 pages)
1. Basic notions
2. Independence
3. Convergence
4. Weak law of large numbers
5. Techniques related to almost sure convergence
6. Strong law of large numbers
7. Uniform integrability
8. Complements to the SLLN
9. Conditional expectation

Contents: (91 pages)
1 Probability spaces
   1.1 Definition of σ-algebras
   1.2 Probability measures
   1.3 Examples of distributions
      1.3.1 Binomial distribution with parameter 0 < p < 1
      1.3.2 Poisson distribution with parameter λ > 0
      1.3.3 Geometric distribution with parameter 0 < p < 1
      1.3.4 Lebesgue measure and uniform distribution
      1.3.5 Gaussian distribution on R with mean m ∈ R and variance σ 2 > 0
      1.3.6 Exponential distribution on R with parameter λ > 0
      1.3.7 Poisson’s Theorem
    1.4 A set which is not a Borel set
2 Random variables
   2.1 Random variables
   2.2 Measurable maps
   2.3 Independence
3 Integration
   3.1 Definition of the expected value
   3.2 Basic properties of the expected value
   3.3 Connections to the Riemann-integral
   3.4 Change of variables in the expected value
   3.5 Fubini’s Theorem
   3.6 Some inequalities
   3.7 Theorem of Radon-Nikodym
   3.8 Modes of convergence
4 Exercises
   4.1 Probability spaces
   4.2 Random variables
   4.3 Integration

Contents: (78 pages)
1 σ-algebras and Borel functions
2 Measures
3 Probability spaces, random variables and distribution functions.
4 Integration theory
5 Expectation in a probability space
6 Characteristic functions
7 Independence
8 Convergence of random variables
9 The strong law of large numbers
10 Stochastic processes