Оглавление
Robert P. Dobrow. Probability
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
PROBABILITY. With Applications and R
PREFACE
ACKNOWLEDGMENTS
ABOUT THE COMPANION WEBSITE
INTRODUCTION
I.1 Walking the Web
I.2 Benford's Law
I.3 Searching the Genome
I.4 Big Data
I.5 From Application to Theory
1 FIRST PRINCIPLES
Learning Outcomes
1.1 RANDOM EXPERIMENT, SAMPLE SPACE, EVENT
1.2 WHAT IS A PROBABILITY?
1.3 PROBABILITY FUNCTION
PROBABILITY FUNCTION
1.4 PROPERTIES OF PROBABILITIES
ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS
EXTENSION OF ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS
PROPERTIES OF PROBABILITIES
1.5 EQUALLY LIKELY OUTCOMES
1.6 COUNTING I
Multiplication principle
1.6.1 Permutations
COUNTING PERMUTATIONS
Sampling with and without replacement
1.7 COUNTING II
1.7.1 Combinations and Binomial Coefficients
COUNTING k-ELEMENT SUBSETS AND LISTS WITH k ONES
Perfect Bridge Hands
Binomial Theorem
1.8 PROBLEM-SOLVING STRATEGIES: COMPLEMENTS AND INCLUSION–EXCLUSION
INCLUSION–EXCLUSION
1.9 A FIRST LOOK AT SIMULATION
MONTE CARLO SIMULATION
R: SIMULATING THE PROBABILITY OF THREE HEADS IN THREE COIN TOSSES
R: SIMULATING THE DIVISIBILITY PROBABILITY
1.10 SUMMARY
EXERCISES. Understanding Sample Spaces and Events
Probability Functions
Equally Likely Outcomes and Counting
Properties of Probabilities
Simulation and R
Chapter Review
2 CONDITIONAL PROBABILITY AND INDEPENDENCE
Learning Outcomes
2.1 CONDITIONAL PROBABILITY
CONDITIONAL PROBABILITY
R: SIMULATING A CONDITIONAL PROBABILITY
2.2 NEW INFORMATION CHANGES THE SAMPLE SPACE
2.3 FINDING P(A AND B)
GENERAL FORMULA FOR P (A AND B)
R: SIMULATING BLACKJACK
2.3.1 Birthday Problem
2.4 CONDITIONING AND THE LAW OF TOTAL PROBABILITY
LAW OF TOTAL PROBABILITY
R: FINDING THE LARGEST NUMBER
R: SIMULATING RANDOM PERMUTATIONS
2.5 BAYES FORMULA AND INVERTING A CONDITIONAL PROBABILITY
Bayes formula
2.6 INDEPENDENCE AND DEPENDENCE
INDEPENDENT EVENTS
COIN TOSSING IN THE REAL WORLD
INDEPENDENCE FOR A COLLECTION OF EVENTS
A BEFORE B
2.7 PRODUCT SPACES*
2.8 SUMMARY
EXERCISES. Basics of Conditional Probability
Conditioning, Law of Total Probability, and Bayes Formula
Independence
Simulation and R
Chapter Review
3 INTRODUCTION TO DISCRETE RANDOM VARIABLES
Learning Outcomes
3.1 RANDOM VARIABLES
RANDOM VARIABLE
UNIFORM RANDOM VARIABLE
3.2 INDEPENDENT RANDOM VARIABLES
INDEPENDENCE OF RANDOM VARIABLES
INDEPENDENT RANDOM VARIABLES
3.3 BERNOULLI SEQUENCES
BERNOULLI DISTRIBUTION
INDEPENDENT AND IDENTICALLY DISTRIBUTED (i.i.d.) SEQUENCES OF RANDOM VARIABLES
3.4 BINOMIAL DISTRIBUTION
BINOMIAL DISTRIBUTION
VISUALIZING THE BINOMIAL DISTRIBUTION
R: WORKING WITH PROBABILITY DISTRIBUTIONS
R: SIMULATING THE OVERBOOKING PROBABILITY
3.5 POISSON DISTRIBUTION
POISSON DISTRIBUTION
R: Poisson distribution
R: SIMULATING ANNUAL ACCIDENT COST
3.5.1 Poisson Approximation of Binomial Distribution
ON THE EDGE
3.5.2 Poisson as Limit of Binomial Probabilities*
3.6 SUMMARY
EXERCISES. Random Variables and Independence
Binomial Distribution
Poisson Distribution
Simulation and R
Chapter Review
4 EXPECTATION AND MORE WITH DISCRETE RANDOM VARIABLES
Learning Outcomes
PROBABILITY MASS FUNCTION
A Note on Notation
4.1 EXPECTATION
EXPECTATION
R: PLAYING ROULETTE
4.2 FUNCTIONS OF RANDOM VARIABLES
R: LEMONADE PROFITS
EXPECTATION OF FUNCTION OF A RANDOM VARIABLE
EXPECTATION OF A LINEAR FUNCTION OF X
4.3 JOINT DISTRIBUTIONS
MARGINAL DISTRIBUTIONS
EXPECTATION OF FUNCTION OF TWO RANDOM VARIABLES
4.4 INDEPENDENT RANDOM VARIABLES
R: DICE AND COINS
FUNCTIONS OF INDEPENDENT RANDOM VARIABLES ARE INDEPENDENT
EXPECTATION OF A PRODUCT OF INDEPENDENT RANDOM VARIABLES
R: EXPECTED VOLUME
4.4.1 Sums of Independent Random Variables
THE SUM OF INDEPENDENT POISSON RANDOM VARIABLES IS POISSON
4.5 LINEARITY OF EXPECTATION
LINEARITY OF EXPECTATION
R: SIMULATING THE MATCHING PROBLEM
4.6 VARIANCE AND STANDARD DEVIATION
VARIANCE AND STANDARD DEVIATION
EXPECTATION AND VARIANCE OF INDICATOR VARIABLE
PROPERTIES OF EXPECTATION, VARIANCE, AND STANDARD DEVIATION
R: SIMULATION OF POISSON DISTRIBUTION
VARIANCE OF THE SUM OF INDEPENDENT VARIABLES
R: A MILLION RED BETS
4.7 COVARIANCE AND CORRELATION
COVARIANCE
CORRELATION
UNCORRELATED RANDOM VARIABLES
GENERAL FORMULA FOR VARIANCE OF A SUM
VARIANCE OF SUM OF N RANDOM VARIABLES
4.8 CONDITIONAL DISTRIBUTION
CONDITIONAL PROBABILITY MASS FUNCTION
4.8.1 Introduction to Conditional Expectation
CONDITIONAL EXPECTATION OF Y GIVEN X = x
R: SIMULATING A CONDITIONAL EXPECTATION
4.9 PROPERTIES OF COVARIANCE AND CORRELATION*
COVARIANCE PROPERTY: LINEARITY
CORRELATION RESULTS
4.10 EXPECTATION OF A FUNCTION OF A RANDOM VARIABLE*
4.11 SUMMARY
EXERCISES. Expectation
Joint Distribution
Variance, Standard Deviation, Covariance, Correlation
Conditional Distribution, Expectation, Functions of Random Variables
Simulation and R
Chapter Review
5 MORE DISCRETE DISTRIBUTIONS AND THEIR RELATIONSHIPS
Learning Outcomes
5.1 GEOMETRIC DISTRIBUTION
GEOMETRIC DISTRIBUTION
TAIL PROBABILITY OF GEOMETRIC DISTRIBUTION
5.1.1 Memorylessness
MEMORYLESS PROPERTY
5.1.2 Coupon Collecting and Tiger Counting
R: NUMERICAL SOLUTION TO TIGER PROBLEM
R: GEOMETRIC DISTRIBUTION
5.2 MOMENT-GENERATING FUNCTIONS
MOMENT-GENERATING FUNCTION
Remarks:
PROPERTIES OF MOMENT GENERATING FUNCTIONS
5.3 NEGATIVE BINOMIAL—UP FROM THE GEOMETRIC
NEGATIVE BINOMIAL DISTRIBUTION
R: NEGATIVE BINOMIAL DISTRIBUTION
5.4 HYPERGEOMETRIC—SAMPLING WITHOUT REPLACEMENT
HYPERGEOMETRIC DISTRIBUTION
R: HYPERGEOMETRIC DISTRIBUTION
R: SIMULATING ACES IN A BRIDGE HAND
5.5 FROM BINOMIAL TO MULTINOMIAL
MULTINOMIAL DISTRIBUTION
MULTINOMIAL THEOREM
R: MULTINOMIAL CALCULATION
5.6 BENFORD'S LAW*
5.7 SUMMARY
EXERCISES. Geometric Distribution
MGFS
Negative Binomial Distribution
Hypergeometric Distribution
Multinomial Distribution
Benford's Law
Other
Simulation and R
Chapter Review
6 CONTINUOUS PROBABILITY
Learning Outcomes
6.1 PROBABILITY DENSITY FUNCTION
PROBABILITY DENSITY FUNCTION
Note: And 0, Otherwise
6.2 CUMULATIVE DISTRIBUTION FUNCTION
CUMULATIVE DISTRIBUTION FUNCTION
CUMULATIVE DISTRIBUTION FUNCTION
6.3 EXPECTATION AND VARIANCE
EXPECTATION AND VARIANCE FOR CONTINUOUS RANDOM VARIABLES
PROPERTIES OF EXPECTATION AND VARIANCE
EXPECTATION OF FUNCTION OF CONTINUOUS RANDOM VARIABLE
6.4 UNIFORM DISTRIBUTION
UNIFORM DISTRIBUTION
R: UNIFORM DISTRIBUTION
R: SIMULATING BALLOON VOLUME
6.5 EXPONENTIAL DISTRIBUTION
EXPONENTIAL DISTRIBUTION
R: EXPONENTIAL DISTRIBUTION
6.5.1 Memorylessness
MEMORYLESSNESS FOR EXPONENTIAL DISTRIBUTION
R: BUS WAITING TIME
6.6 JOINT DISTRIBUTIONS
JOINT DENSITY FUNCTION
JOINT CUMULATIVE DISTRIBUTION FUNCTION
UNIFORM DISTRIBUTION IN TWO DIMENSIONS
MARGINAL DISTRIBUTIONS FROM JOINT DENSITIES
EXPECTATION OF FUNCTION OF JOINTLY DISTRIBUTED RANDOM VARIABLES
6.7 INDEPENDENCE
INDEPENDENCE AND DENSITY FUNCTIONS
6.7.1 Accept–Reject Method
6.8 COVARIANCE, CORRELATION
COVARIANCE
R: SIMULATION OF COVARIANCE, CORRELATION
6.9 SUMMARY
EXERCISES. Density, cdf, Expectation, Variance
Exponential Distribution
Joint Distributions, Independence, Covariance
Simulation and R
Chapter Review
7 CONTINUOUS DISTRIBUTIONS
Learning Outcomes
7.1 NORMAL DISTRIBUTION
NORMAL DISTRIBUTION
R: NORMAL DISTRIBUTION
7.1.1 Standard Normal Distribution
LINEAR FUNCTION OF NORMAL RANDOM VARIABLE
7.1.2 Normal Approximation of Binomial Distribution
7.1.3 Quantiles
QUANTILE
10,000 COIN FLIPS
7.1.4 Sums of Independent Normals
SUM OF INDEPENDENT NORMAL RANDOM VARIABLES IS NORMAL
AVERAGES OF i.i.d. RANDOM VARIABLES
7.2 GAMMA DISTRIBUTION
GAMMA DISTRIBUTION
SUM OF EXPONENTIALS HAS GAMMA DISTRIBUTION
R: SIMULATING THE GAMMA DISTRIBUTION FROM A SUM OF EXPONENTIALS
7.2.1 Probability as a Technique of Integration
7.3 POISSON PROCESS
DISTRIBUTION OF FOR POISSON PROCESS WITH PARAMETER
PROPERTIES OF POISSON PROCESS
R: SIMULATING A POISSON PROCESS
7.4 BETA DISTRIBUTION
7.5 PARETO DISTRIBUTION*
PARETO DISTRIBUTION
SCALE-INVARIANCE
7.6 SUMMARY
EXERCISES. Normal Distribution
Gamma Distribution, Poisson Process
Beta Distribution
Pareto, Scale-invariant Distribution
Simulation and R
Chapter Review
8 DENSITIES OF FUNCTIONS OF RANDOM VARIABLES
Learning Outcomes
8.1 DENSITIES VIA CDFs
R: COMPARING THE EXACT DISTRIBUTION WITH A SIMULATION
HOW TO FIND THE DENSITY OF Y = g(X)
8.1.1 Simulating a Continuous Random Variable
INVERSE TRANSFORM METHOD
R: IMPLEMENTING THE INVERSE TRANSFORM METHOD
8.1.2 Method of Transformations
8.2 MAXIMUMS, MINIMUMS, AND ORDER STATISTICS
INEQUALITIES FOR MAXIMUMS AND MINIMUMS
MINIMUM OF INDEPENDENT EXPONENTIAL DISTRIBUTIONS
8.3 CONVOLUTION
8.4 GEOMETRIC PROBABILITY
Ants, Fish, and Noodles
8.5 TRANSFORMATIONS OF TWO RANDOM VARIABLES*
JOINT DENSITY OF and
8.6 SUMMARY
EXERCISES. Practice with Finding Densities
Maxs, Mins, and Convolution
Geometric Probability
Bivariate Transformations
Simulation and R
Chapter Review
9 CONDITIONAL DISTRIBUTION, EXPECTATION, AND VARIANCE
Learning Outcomes
INTRODUCTION
9.1 CONDITIONAL DISTRIBUTIONS
CONDITIONAL DENSITY FUNCTION
BAYES FORMULA
9.2 DISCRETE AND CONTINUOUS: MIXING IT UP
R: SIMULATING EXPONENTIAL-POISSON TRAFFIC FLOW MODEL
9.3 CONDITIONAL EXPECTATION
CONDITIONAL EXPECTATION OF GIVEN
9.3.1 From Function to Random Variable
CONDITIONAL EXPECTATION E[ Y|X]
LAW OF TOTAL EXPECTATION
PROPERTIES OF CONDITIONAL EXPECTATION
9.3.2 Random Sum of Random Variables
9.4 COMPUTING PROBABILITIES BY CONDITIONING
9.5 CONDITIONAL VARIANCE
CONDITIONAL VARIANCE OF GIVEN
PROPERTIES OF CONDITIONAL VARIANCE
LAW OF TOTAL VARIANCE
R: TOTAL SPENDING AT ALICE'S RESTAURANT
9.6 BIVARIATE NORMAL DISTRIBUTION*
BIVARIATE STANDARD NORMAL DISTRIBUTION
BIVARIATE NORMAL DENSITY
PROPERTIES OF BIVARIATE STANDARD NORMAL DISTRIBUTION
R: SIMULATING BIVARIATE NORMAL RANDOM VARIABLES
CONDITIONAL DISTRIBUTION OF GIVEN
9.7 SUMMARY
EXERCISES. Conditional Distributions
Conditional Expectation
Computing Probabilities with Conditioning
Conditional Variance
Bivariate Normal Distribution
Simulation and R
Chapter Review
10 LIMITS
Learning Outcomes
THE “LAW OF AVERAGES” AND A RUN OF BLACK AT THE CASINO
10.1 WEAK LAW OF LARGE NUMBERS
R: WEAK LAW OF LARGE NUMBERS
10.1.1 Markov and Chebyshev Inequalities
WEAK LAW OF LARGE NUMBERS
Remarks
10.2 STRONG LAW OF LARGE NUMBERS
STRONG LAW OF LARGE NUMBERS
Remarks
10.3 METHOD OF MOMENTS*
10.4 MONTE CARLO INTEGRATION
MONTE CARLO INTEGRATION ON
10.5 CENTRAL LIMIT THEOREM
CENTRAL LIMIT THEOREM (CLT)
R: SIMULATION EXPERIMENT
Remarks
R: RANDOM WALK DISTANCE FROM ORIGIN
10.5.1 Central Limit Theorem and Monte Carlo
10.6 A PROOF OF THE CENTRAL LIMIT THEOREM
CONTINUITY THEOREM
10.7 SUMMARY
EXERCISES. Law of Large Numbers
Applications: Method of Moments and Monte Carlo Integration
Central Limit Theorem
Simulation and R
Chapter Review
11 BEYOND RANDOM WALKS AND MARKOV CHAINS
Learning Outcomes
11.1 RANDOM WALKS ON GRAPHS
11.1.1 Long-Term Behavior
LIMITING DISTRIBUTION
LIMITING DISTRIBUTION FOR A SIMPLE RANDOM WALK ON A GRAPH
R: RANDOM WALK ON A GRAPH
Remarks
11.2 RANDOM WALKS ON WEIGHTED GRAPHS AND MARKOV CHAINS
11.2.1 Stationary Distribution
STATIONARY DISTRIBUTION
STATIONARY, LIMITING DISTRIBUTION FOR RANDOM WALK ON WEIGHTED GRAPHS
DETAILED BALANCE CONDITION
11.3 FROM MARKOV CHAIN TO MARKOV CHAIN MONTE CARLO
MCMC: METROPOLIS–HASTINGS ALGORITHM
R: MCMC—A TOY EXAMPLE
Remarks
R: SIMULATION OF BIVARIATE STANDARD NORMAL DISTRIBUTION
11.4 SUMMARY
EXERCISES. Random Walk on Graphs and Markov Chains
Markov Chain Monte Carlo
Chapter Review
APPENDIX A PROBABILITY DISTRIBUTIONS IN R
APPENDIX B SUMMARY OF PROBABILITY DISTRIBUTIONS
APPENDIX C MATHEMATICAL REMINDERS. Exponents
Logarithms
Calculus
Series
APPENDIX D WORKING WITH JOINT DISTRIBUTIONS
SOLUTIONS TO EXERCISES. SOLUTIONS FOR CHAPTER 1
SOLUTIONS FOR CHAPTER 2
SOLUTIONS FOR CHAPTER 3
SOLUTIONS FOR CHAPTER 4
SOLUTIONS FOR CHAPTER 5
SOLUTIONS FOR CHAPTER 6
SOLUTIONS FOR CHAPTER 7
SOLUTIONS FOR CHAPTER 8
SOLUTIONS FOR CHAPTER 9
SOLUTIONS FOR CHAPTER 10
SOLUTIONS FOR CHAPTER 11
REFERENCES
INDEX
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