Probability and Statistics for Computer Scientists

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
1. Introduction and Overview
	1.1 Making decisions under uncertainty
	1.2 Overview of this book
	Summary and conclusions
	Exercises
Part I: Probability and Random Variables
	2. Probability
		2.1 Events and their probabilities
			2.1.1 Outcomes, events, and the sample space
			2.1.2 Set operations
		2.2 Rules of Probability
			2.2.1 Axioms of Probability
			2.2.2 Computing probabilities of events
			2.2.3 Applications in reliability
		2.3 Combinatorics
			2.3.1 Equally likely outcomes
			2.3.2 Permutations and combinations
		2.4 Conditional probability and independence
		Summary and conclusions
		Exercises
	3. Discrete Random Variables and Their Distributions
		3.1 Distribution of a random variable
			3.1.1 Main concepts
			3.1.2 Types of random variables
		3.2 Distribution of a random vector
			3.2.1 Joint distribution and marginal distributions
			3.2.2 Independence of random variables
		3.3 Expectation and variance
			3.3.1 Expectation
			3.3.2 Expectation of a function
			3.3.3 Properties
			3.3.4 Variance and standard deviation
			3.3.5 Covariance and correlation
			3.3.6 Properties
			3.3.7 Chebyshev’s inequality
			3.3.8 Application to finance
		3.4 Families of discrete distributions
			3.4.1 Bernoulli distribution
			3.4.2 Binomial distribution
			3.4.3 Geometric distribution
			3.4.4 Negative Binomial distribution
			3.4.5 Poisson distribution
			3.4.6 Poisson approximation of Binomial distribution
		Summary and conclusions
		Exercises
	4. Continuous Distributions
		4.1 Probability density
		4.2 Families of continuous distributions
			4.2.1 Uniform distribution
			4.2.2 Exponential distribution
			4.2.3 Gamma distribution
			4.2.4 Normal distribution
		4.3 Central Limit Theorem
		Summary and conclusions
		Exercises
	5. Computer Simulations and Monte Carlo Methods
		5.1 Introduction
			5.1.1 Applications and examples
		5.2 Simulation of random variables
			5.2.1 Random number generators
			5.2.2 Discrete methods
			5.2.3 Inverse transform method
			5.2.4 Rejection method
			5.2.5 Generation of random vectors
			5.2.6 Special methods
		5.3 Solving problems by Monte Carlo methods
			5.3.1 Estimating probabilities
			5.3.2 Estimating means and standard deviations
			5.3.3 Forecasting
			5.3.4 Estimating lengths, areas, and volumes
			5.3.5 Monte Carlo integration
		Summary and conclusions
		Exercises
Part II: Stochastic Processes
	6. Stochastic Processes
		6.1 Definitions and classifications
		6.2 Markov processes and Markov chains
			6.2.1 Markov chains
			6.2.2 Matrix approach
			6.2.3 Steady-state distribution
		6.3 Counting processes
			6.3.1 Binomial process
			6.3.2 Poisson process
		6.4 Simulation of stochastic processes
		Summary and conclusions
		Exercises
	7. Queuing Systems
		7.1 Main components of a queuing system
		7.2 The Little’s Law
		7.3 Bernoulli single-server queuing process
			7.3.1 Systems with limited capacity
		7.4 M/M/1 system
			7.4.1 Evaluating the system’s performance
		7.5 Multiserver queuing systems
			7.5.1 Bernoulli K-server queuing process
			7.5.2 M/M/k systems
			7.5.3 Unlimited number of servers and M/M/8 ∞
		7.6 Simulation of queuing systems
		Summary and conclusions
		Exercises
Part III: Statistics
	8. Introduction to Statistics
		8.1 Population and sample, parameters and statistics
		8.2 Descriptive statistics
			8.2.1 Mean
			8.2.2 Median
			8.2.3 Quantiles, percentiles, and quartiles
			8.2.4 Variance and standard deviation
			8.2.5 Standard errors of estimates
			8.2.6 Interquartile range
		8.3 Graphical statistics
			8.3.1 Histogram
			8.3.2 Stem-and-leaf plot
			8.3.3 Boxplot
			8.3.4 Scatter plots and time plots
		Summary and conclusions
		Exercises
	9. Statistical Inference I
		9.1 Parameter estimation
			9.1.1 Method of moments
			9.1.2 Method of maximum likelihood
			9.1.3 Estimation of standard errors
		9.2 Confidence intervals
			9.2.1 Construction of confidence intervals: a general method
			9.2.2 Confidence interval for the population mean
			9.2.3 Confidence interval for the difference between two means
			9.2.4 Selection of a sample size
			9.2.5 Estimating means with a given precision
		9.3 Unknown standard deviation
			9.3.1 Large samples
			9.3.2 Confidence intervals for proportions
			9.3.3 Estimating proportions with a given precision
			9.3.4 Small samples: Student’s t distribution
			9.3.5 Comparison of two populations with unknown variances
		9.4 Hypothesis testing
			9.4.1 Hypothesis and alternative
			9.4.2 Type I and Type II errors: level of significance
			9.4.3 Level a tests: general approach
			9.4.4 Rejection regions and power
			9.4.5 Standard Normal null distribution (Z-test)
			9.4.6 Z-tests for means and proportions
			9.4.7 Pooled sample proportion
			9.4.8 Unknown σs: T-tests
			9.4.9 Duality: two-sided tests and two-sided confidence intervals
			9.4.10 P-value
		9.5 Inference about variances
			9.5.1 Variance estimator and Chi-square distribution
			9.5.2 Confidence interval for the population variance
			9.5.3 Testing variance
			9.5.4 Comparison of two variances. F-distribution
			9.5.5 Confidence interval for the ratio of population variances
			9.5.6 F-tests comparing two variances
		Summary and conclusions
		Exercises
	10. Statistical Inference II
		10.1 Chi-square tests
			10.1.1 Testing a distribution
			10.1.2 Testing a family of distributions
			10.1.3 Testing independence
		10.2 Nonparametric statistics
			10.2.1 Sign test
			10.2.2 Wilcoxon signed rank test
			10.2.3 Mann–Whitney–Wilcoxon rank sum test
		10.3 Bootstrap
			10.3.1 Bootstrap distribution and all bootstrap samples
			10.3.2 Computer generated bootstrap samples
			10.3.3 Bootstrap confidence intervals
		10.4 Bayesian inference
			10.4.1 Prior and posterior
			10.4.2 Bayesian estimation
			10.4.3 Bayesian credible sets
			10.4.4 Bayesian hypothesis testing
		Summary and conclusions
		Exercises
	11. Regression
		11.1 Least squares estimation
			11.1.1 Examples
			11.1.2 Method of least squares
			11.1.3 Linear regression
			11.1.4 Regression and correlation
			11.1.5 Overfitting a model
		11.2 Analysis of variance, prediction, and further inference
			11.2.1 ANOVA and R-square
			11.2.2 Tests and confidence intervals
			11.2.3 Prediction
		11.3 Multivariate regression
			11.3.1 Introduction and examples
			11.3.2 Matrix approach and least squares estimation
			11.3.3 Analysis of variance, tests, and prediction
		11.4 Model building
			11.4.1 Adjusted R-square
			11.4.2 Extra sum of squares, partial F-tests, and variable selection
			11.4.3 Categorical predictors and dummy variables
		Summary and conclusions
		Exercises
Appendix
	A.1 Data sets
	A.2 Inventory of distributions
		A.2.1 Discrete families
		A.2.2 Continuous families
	A.3 Distribution tables
	A.4 Calculus review
		A.4.1 Inverse function
		A.4.2 Limits and continuity
		A.4.3 Sequences and series
		A.4.4 Derivatives, minimum, and maximum
		A.4.5 Integrals
	A.5 Matrices and linear systems
	A.6 Answers to selected exercises
Index