Probability and statistics for computer scientists 2nd Edition by Michael Baron 1482214105 9781482214109

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ISBN 10: 1482214105 

ISBN 13: 9781482214109

Author: Michael Baron

Table of contents:

Chapter 1 Introduction and Overview
1.1 Making decisions under uncertainty
Summary and conclusion
1.2 Overview of this book
Summary and conclusions
Exercises
Figure 1.1

Part I Probability and Random Variables

Chapter 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
Extreme cases
Union
Complement
Intersection of independent events
2.2.3 Applications in reliability
2.3 Combinatorics
2.3.1 Equally likely outcomes
2.3.2 Permutations and combinations
Permutations with replacement
Permutations without replacement
Combinations without replacement
Computational shortcuts
Combinations with replacement
2.4 Conditional probability and independence
Conditional probability
Independence
Bayes Rule
Law of Total Probability
Summary and conclusions
Exercises
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8

Chapter 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
MATLAB notes
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
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6

Chapter 4 Continuous Distributions
4.1 Probability density
Analogy: pmf versus pdf
Joint and marginal densities
Expectation and variance
4.2 Families of continuous distributions
4.2.1 Uniform distribution
The Uniform property
Standard Uniform distribution
Expectation and variance
4.2.2 Exponential distribution
Times between rare events are Exponential
Memoryless property
4.2.3 Gamma distribution
Expectation, variance, and some useful integration remarks
Gamma-Poisson formula
4.2.4 Normal distribution
Standard Normal distribution
4.3 Central Limit Theorem
Normal approximation to Binomial distribution
Continuity correction
Summary and conclusions
Exercises
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Table 4.1
Table 4.2
Table 4.3

Chapter 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
Arbitrary discrete distribution
5.2.3 Inverse transform method
Arbitrary continuous distribution
Discrete distributions revisited
Exponential-Geometric relation
5.2.4 Rejection method
5.2.5 Generation of random vectors
5.2.6 Special methods
Poisson distribution
Normal distribution
5.3 Solving problems by Monte Carlo methods
5.3.1 Estimating probabilities
Accuracy of a Monte Carlo study
5.3.2 Estimating means and standard deviations
5.3.3 Forecasting
5.3.4 Estimating lengths, areas, and volumes
Lengths
Areas and volumes
Areas of arbitrary regions with unknown boundaries
5.3.5 Monte Carlo integration
Accuracy of results
Improved Monte Carlo integration method
Accuracy of the improved method
Summary and conclusions
Exercises
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8

Part II Stochastic Processes

Chapter 6 Stochastic Processes
6.1 Definitions and classifications
6.2 Markov processes and Markov chains
6.2.1 Markov chains
Characteristics of a Markov chain
6.2.2 Matrix approach
6.2.3 Steady-state distribution
Computing the steady-state distribution
The limit of Ph
Steady state
Existence of a steady-state distribution. Regular Markov chains
Conclusion
6.3 Counting processes
6.3.1 Binomial process
Relation to real time: frames
Markov property
6.3.2 Poisson process
Continuous time
Poisson process as the limiting case
Rare events and modeling
6.4 Simulation of stochastic processes
Discrete-time processes
Markov chains
Binomial process
Continuous-time processes
Poisson process
Summary and conclusions
Exercises
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 6.7
Figure 6.8
Figure 6.9
Figure 6.10

Chapter 7 Queuing Systems
7.1 Main components of a queuing system
Arrival
Queuing and routing to servers
Service
Departure
7.2 The Little’s Law
7.3 Bernoulli single-server queuing process
Markov property
Steady-state distribution
7.3.1 Systems with limited capacity
7.4 M/M/1 system
M/M/1 as a limiting case of a Bernoulli queuing process
Steady-state distribution for an M/M/1 system
7.4.1 Evaluating the system’s performance
Utilization
Waiting time
Response time
Queue
Little’s Law revisited
When a system gets nearly overloaded
7.5 Multiserver queuing systems
7.5.1 Bernoulli k-server queuing process
Markov property
Transition probabilities
7.5.2 M/M/k systems
Steady-state distribution
7.5.3 Unlimited number of servers and M/M/∞
M/M/∞ queueing system
7.6 Simulation of queuing systems
Markov case
General case
Example: simulation of a multiserver queuing system
Summary and conclusions
Exercises
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5

Part III Statistics

Chapter 8 Introduction to Statistics
8.1 Population and sample, parameters and statistics
Sampling and non-sampling errors
8.2 Simple descriptive statistics
8.2.1 Mean
Unbiasedness
Consistency
Asymptotic Normality
8.2.2 Median
Understanding the shape of a distribution
Computation of a population median
Computing sample medians
8.2.3 Quantiles, percentiles, and quartiles
8.2.4 Variance and standard deviation
Computation
8.2.5 Standard errors of estimates
8.2.6 Interquartile range
Detection of outliers
Handling of outliers
8.3 Graphical statistics
8.3.1 Histogram
How else may histograms look like?
Mixtures
The choice of bins
8.3.2 Stem-and-leaf plot
8.3.3 Boxplot
Parallel boxplots
8.3.4 Scatter plots and time plots
MATLAB notes
Summary and conclusions
Exercises
Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5
Figure 8.6
Figure 8.7
Figure 8.8
Figure 8.9
Figure 8.10
Figure 8.11
Figure 8.12

Chapter 9 Statistical Inference I
9.1 Parameter estimation
9.1.1 Method of moments
Moments
Estimation
9.1.2 Method of maximum likelihood
Discrete case
Continuous case
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
Case 1. Equal 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 α tests: general approach
Step 1. Test statistic
Step 2. Acceptance region and rejection region
Step 3: Result and its interpretation
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 σ: T-tests
9.4.9 Duality: two-sided tests and two-sided confidence intervals
9.4.10 P-value
How do we choose α?
P-value
Testing hypotheses with a P-value
Computing P-values
Understanding P-values
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
Level α test
P-value
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
MATLAB notes
Summary and conclusions
Exercises
Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4
Figure 9.5
Figure 9.6
Figure 9.7
Figure 9.8
Figure 9.9

Figure 9.10 
Figure 9.11 
Figure 9.12 
Figure 9.13 
Figure 9.14 
Figure 9.15 
Table 9.1 
Table 9.2 
Table 9.3 
Table 9.4 
Table 9.5 
Table 9.6 
Chapter 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 
Null distribution of Wilcoxon test statistic 
Exact distribution 
Normal approximation 
10.2.3 Mann-Whitney-Wilcoxon rank sum test 
Mann-Whitney-Wilcoxon test in MATLAB 
Null distribution of Mann-Whitney-Wilcoxon test statistic 
Normal approximation 
10.3 Bootstrap 
10.3.1 Bootstrap distribution and all bootstrap samples 
Bootstrap distribution 
10.3.2 Computer generated bootstrap samples 
10.3.3 Bootstrap confidence intervals 
Parametric method, based on the bootstrap estimation of the standard error 
Nonparametric method, based on the bootstrap quantiles 
10.4 Bayesian inference 
10.4.1 Prior and posterior 
Conjugate distribution families 
Gamma family is conjugate to the Poisson model 
Beta family is conjugate to the Binomial model 
Normal family is conjugate to the Normal model 
10.4.2 Bayesian estimation 
10.4.3 Bayesian credible sets 
10.4.4 Bayesian hypothesis testing 
Loss and risk 
Summary and conclusions 
Exercises 
Figure 10.1 
Figure 10.2 
Figure 10.3 
Figure 10.4 
Figure 10.5 
Figure 10.6 
Figure 10.7 
Figure 10.8 
Figure 10.9 
Figure 10.10 
Figure 10.11 
Figure 10.12 
Figure 10.13 
Figure 10.14 
Table 10.1 
Table 10.2 
Chapter 11 Regression 
11.1 Least squares estimation 
11.1.1 Examples 
11.1.2 Method of least squares 
11.1.3 Linear regression 
Estimation in 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 
Degrees of freedom and variance estimation 
Inference about the regression slope 
ANOVA F-test 
F-test and T-test 
11.2.3 Prediction 
Confidence interval for the mean of responses 
Prediction interval for the individual response 
Prediction bands 
11.3 Multivariate regression 
11.3.1 Introduction and examples 
11.3.2 Matrix approach and least squares estimation 
Matrix approach to multivariate linear regression 
Least squares estimates 
11.3.3 Analysis of variance, tests, and prediction 
Testing significance of the entire model 
Variance estimator 
Testing individual slopes 
Prediction 
11.4 Model building 
11.4.1 Adjusted R-square 
11.4.2 Extra sum of squares, partial F-tests, and variable selection 
Stepwise (forward) selection 
Backward elimination 
11.4.3 Categorical predictors and dummy variables 
Avoid singularity by creating only (C − 1) dummies 
Interpretation of slopes for dummy variables 
Matlab notes 
Summary and conclusions 
Exercises 
Figure 11.1 
Figure 11.2 
Figure 11.3 
Figure 11.4 
Figure 11.5 
Figure 11.6 
Table 11.1 
Part IV Appendix 
Chapter 12 Appendix 
12.1 Inventory of distributions 
12.1.1 Discrete families 
12.1.2 Continuous families 
12.2 Distribution tables 
12.3 Calculus review 
12.3.1 Inverse function 
12.3.2 Limits and continuity 
12.3.3 Sequences and series 
12.3.4 Derivatives, minimum, and maximum 
Computing maxima and minima 
12.3.5 Integrals 
Improper integrals 
Integration by substitution 
Integration by parts 
Computing areas 
Gamma function and factorial 
12.4 Matrices and linear systems 
Multiplying a row by a column 
Multiplying matrices 
Transposition 
Solving systems of equations 
Inverse matrix 
Matrix operations in Matlab 
12.5 Answers to selected exercises

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