Chapter
3.2.6 Straights, Flushes, and Straight Flushes
3.2.7 Full House and Four of a Kind
3.2.8 Practice Poker Hand: I
3.2.9 Practice Poker Hand: II
3.4.2 Number of Bridge Deals
3.5 Appendix: Coding to Compute Probabilities
3.5.1 Trump Split and Code
4 Conditional Probability, Independence, and Bayes’ Theorem
4.1 Conditional Probabilities
4.1.1 Guessing the Conditional Probability Formula
4.1.2 Expected Counts Approach
4.1.3 Venn Diagram Approach
4.1.4 The Monty Hall Problem
4.2 The General Multiplication Rule
4.2.3 Hat Problem and Error Correcting Codes
4.2.4 Advanced Remark: Definition of Conditional Probability
4.5 Partitions and the Law of Total Probability
4.6 Bayes’ Theorem Revisited
5 Counting II: Inclusion-Exclusion
5.1 Factorial and Binomial Problems
5.1.1 “How many” versus “What’s the probability”
5.2 The Method of Inclusion-Exclusion
5.2.1 Special Cases of the Inclusion-Exclusion Principle
5.2.2 Statement of the Inclusion-Exclusion Principle
5.2.3 Justification of the Inclusion-Exclusion Formula
5.2.4 Using Inclusion-Exclusion: Suited Hand
5.2.5 The At Least to Exactly Method
5.3.1 Counting Derangements
5.3.2 The Probability of a Derangement
5.3.3 Coding Derangement Experiments
5.3.4 Applications of Derangements
6 Counting III: Advanced Combinatorics
6.1.1 Enumerating Cases: I
6.1.2 Enumerating Cases: II
6.1.3 Sampling With and Without Replacement
6.2.2 Multinomial Coefficients
6.3.3 Additional Partitions
II Introduction to Random Variables
7 Introduction to Discrete Random Variables
7.1 Discrete Random Variables: Definition
7.2 Discrete Random Variables: PDFs
7.3 Discrete Random Variables: CDFs
8 Introduction to Continuous Random Variables
8.1 Fundamental Theorem of Calculus
8.2 PDFs and CDFs: Definitions
8.3 PDFs and CDFs: Examples
8.4 Probabilities of Singleton Events
9.2 Expected Values and Moments
9.5 Linearity of Expectation
9.6 Properties of the Mean and the Variance
9.7 Skewness and Kurtosis
10 Tools: Convolutions and Changing Variables
10.1 Convolutions: Definitions and Properties
10.2 Convolutions: Die Example
10.2.1 Theoretical Calculation
10.3 Convolutions of Several Variables
10.4 Change of Variable Formula: Statement
10.5 Change of Variables Formula: Proof
10.6 Appendix: Products and Quotients of Random Variables
10.6.1 Density of a Product
10.6.2 Density of a Quotient
10.6.3 Example: Quotient of Exponentials
11 Tools: Differentiating Identities
11.1 Geometric Series Example
11.2 Method of Differentiating Identities
11.3 Applications to Binomial Random Variables
11.4 Applications to Normal Random Variables
11.5 Applications to Exponential Random Variables
III Special Distributions
12 Discrete Distributions
12.1 The Bernoulli Distribution
12.2 The Binomial Distribution
12.3 The Multinomial Distribution
12.4 The Geometric Distribution
12.5 The Negative Binomial Distribution
12.6 The Poisson Distribution
12.7 The Discrete Uniform Distribution
13 Continuous Random Variables: Uniform and Exponential
13.1 The Uniform Distribution
13.1.2 Sums of Uniform Random Variables
13.1.4 Generating Random Numbers Uniformly
13.2 The Exponential Distribution
13.2.2 Sums of Exponential Random Variables
13.2.3 Examples and Applications of Exponential Random Variables
13.2.4 Generating Random Numbers from Exponential Distributions
14 Continuous Random Variables: The Normal Distribution
14.1 Determining the Normalization Constant
14.3 Sums of Normal Random Variables
14.3.1 Case 1: μX = µY = 0 and σ2X = σ2Y = 1
14.3.2 Case 2: General µX, µY and σ2X, σ2Y
14.3.3 Sums of Two Normals: Faster Algebra
14.4 Generating Random Numbers from Normal Distributions
14.5 Examples and the Central Limit Theorem
15 The Gamma Function and Related Distributions
15.2 The Functional Equation of Γ (s)
15.3 The Factorial Function and Γ (s)
15.4 Special Values of Γ (s)
15.5 The Beta Function and the Gamma Function
15.5.1 Proof of the Fundamental Relation
15.5.2 The Fundamental Relation and Γ (1/2)
15.6 The Normal Distribution and the Gamma Function
15.7 Families of Random Variables
15.8 Appendix: Cosecant Identity Proofs
15.8.1 The Cosecant Identity: First Proof
15.8.2 The Cosecant Identity: Second Proof
15.8.3 The Cosecant Identity: Special Case s=1/2
16 The Chi-square Distribution
16.1 Origin of the Chi-square Distribution
16.2 Mean and Variance of X ~χ2(1)
16.3 Chi-square Distributions and Sums of Normal Random Variables
16.3.1 Sums of Squares by Direct Integration
16.3.2 Sums of Squares by the Change of Variables Theorem
16.3.3 Sums of Squares by Convolution
16.3.4 Sums of Chi-square Random Variables
17 Inequalities and Laws of Large Numbers
17.3 Chebyshev’s Inequality
17.3.3 Normal and Uniform Examples
17.3.4 Exponential Example
17.4 The Boole and Bonferroni Inequalities
17.5 Types of Convergence
17.5.1 Convergence in Distribution
17.5.2 Convergence in Probability
17.5.3 Almost Sure and Sure Convergence
17.6 Weak and Strong Laws of Large Numbers
18.1 Stirling’s Formula and Probabilities
18.2 Stirling’s Formula and Convergence of Series
18.3 From Stirling to the Central Limit Theorem
18.4 Integral Test and the Poor Man’s Stirling
18.5 Elementary Approaches towards Stirling’s Formula
18.5.1 Dyadic Decompositions
18.5.2 Lower Bounds towards Stirling: I
18.5.3 Lower Bounds toward Stirling II
18.5.4 Lower Bounds towards Stirling: III
18.6 Stationary Phase and Stirling
18.7 The Central Limit Theorem and Stirling
19 Generating Functions and Convolutions
19.3 Uniqueness and Convergence of Generating Functions
19.4 Convolutions I: Discrete Random Variables
19.5 Convolutions II: Continuous Random Variables
19.6 Definition and Properties of Moment Generating Functions
19.7 Applications of Moment Generating Functions
20 Proof of the Central Limit Theorem
20.1 Key Ideas of the Proof
20.2 Statement of the Central Limit Theorem
20.3 Means, Variances, and Standard Deviations
20.5 Needed Moment Generating Function Results
20.6 Special Case: Sums of Poisson Random Variables
20.7 Proof of the CLT for General Sums via MGF
20.8 Using the Central Limit Theorem
20.9 The Central Limit Theorem and Monte Carlo Integration
21 Fourier Analysis and the Central Limit Theorem
21.2 Convolutions and Probability Theory
21.3 Proof of the Central Limit Theorem
22.1.1 Null and Alternative Hypotheses
22.1.2 Significance Levels
22.1.4 One-sided versus Two-sided Tests
22.2.1 Extraordinary Claims and p-values
22.2.3 Misconceptions about p-values
22.3.1 Estimating the Sample Variance
22.3.2 From z-tests to t-tests
22.4 Problems with Hypothesis Testing
22.4.3 Error Rates and the Justice System
22.5 Chi-square Distributions, Goodness of Fit
22.5.1 Chi-square Distributions and Tests of Variance
22.5.2 Chi-square Distributions and t-distributions
22.5.3 Goodness of Fit for List Data
22.6.1 Two-sample z-test: Known Variances
22.6.2 Two-sample t-test: Unknown but Same Variances
22.6.3 Unknown and Different Variances
23 Difference Equations, Markov Processes, and Probability
23.1 From the Fibonacci Numbers to Roulette
23.1.1 The Double-plus-one Strategy
23.1.2 A Quick Review of the Fibonacci Numbers
23.1.3 Recurrence Relations and Probability
23.1.4 Discussion and Generalizations
23.1.5 Code for Roulette Problem
23.2 General Theory of Recurrence Relations
23.2.2 The Characteristic Equation
23.2.3 The Initial Conditions
23.2.4 Proof that Distinct Roots Imply Invertibility
23.3.1 Recurrence Relations and Population Dynamics
23.3.2 General Markov Processes
24 The Method of Least Squares
24.1 Description of the Problem
24.2 Probability and Statistics Review
24.3 The Method of Least Squares
25 Two Famous Problems and Some Coding
25.1 The Marriage/Secretary Problem
25.1.1 Assumptions and Strategy
25.1.2 Probability of Success
25.1.3 Coding the Secretary Problem
25.2.3 Coding the Monty Hall Problem
25.3.1 Sampling with and without Replacement
Appendix A Proof Techniques
A.2.3 The Binomial Theorem
A.2.4 Fibonacci Numbers Modulo 2
A.2.5 False Proofs by Induction
A.4 Proof by Exploiting Symmetries
A.6 Proof by Comparison or Story
A.7 Proof by Contradiction
A.8 Proof by Exhaustion (or Divide and Conquer)
A.9 Proof by Counterexample
A.10 Proof by Generalizing Example
A.11 Dirichlet’s Pigeon-Hole Principle
A.12 Proof by Adding Zero or Multiplying by One
Appendix B Analysis Results
B.1 The Intermediate and Mean Value Theorems
B.2 Interchanging Limits, Derivatives, and Integrals
B.2.1 Interchanging Orders: Theorems
B.2.2 Interchanging Orders: Examples
B.3 Convergence Tests for Series
B.5 The Exponential Function
B.6 Proof of the Cauchy-Schwarz Inequality
Appendix C Countable and Uncountable Sets
C.4 Length of the Rationals
C.5 Length of the Cantor Set
Appendix D Complex Analysis and the Central Limit Theorem
D.1 Warnings from Real Analysis
D.2 Complex Analysis and Topology Definitions
D.3 Complex Analysis and Moment Generating Functions
The Probability Lifesaver
：All the Tools You Need to Understand Chance
( Princeton Lifesaver Study Guides )
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