Probability and Statistics for Computer Scientists, Third Edition

3rd Edition

Chapman and Hall/CRC

491 pages

Hardback: 9781138044487
pub: 2019-06-28
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Description

Probability and statistical methods, simulation techniques, and modeling tools. This third edition textbook adds R, including codes for data analysis examples, helps students solve problems, make optimal decisions in select stochastic models, probabilities and forecasts, and evaluate performance of computer systems and networks.

1. Introduction and Overview

Making decisions under uncertainty

Overview of this book

Summary and conclusions

Exercises

I Probability and Random Variables

2. Probability

Events and their probabilities

Outcomes, events, and the sample space

Set operations

Rules of Probability

Axioms of Probability

Computing probabilities of events

Applications in reliability

Combinatorics

Equally likely outcomes

Permutations and combinations

Conditional probability and independence

Summary and conclusions

Exercises

3. Discrete Random Variables and Their Distributions

Distribution of a random variable

Main concepts

Types of random variables

Distribution of a random vector

Joint distribution and marginal distributions

Independence of random variables

Expectation and variance

Expectation

Expectation of a function

Properties

Variance and standard deviation

Covariance and correlation

Properties

Chebyshev’s inequality

Application to finance

Families of discrete distributions

Bernoulli distribution

Binomial distribution

Geometric distribution

Negative Binomial distribution

Poisson distribution

Poisson approximation of Binomial distribution

Summary and conclusions

Exercises

4. Continuous Distributions

Probability density

Families of continuous distributions

Uniform distribution

Exponential distribution

Gamma distribution

Normal distribution

Central Limit Theorem

Summary and conclusions

Exercises

5. Computer Simulations and Monte Carlo Methods

Introduction

Applications and examples

Simulation of random variables

Random number generators

Discrete methods

Inverse transform method

Rejection method

Generation of random vectors

Special methods

Solving problems by Monte Carlo methods

Estimating probabilities

Estimating means and standard deviations

Forecasting

Estimating lengths, areas, and volumes

Monte Carlo integration

Summary and conclusions

Exercises

II Stochastic Processes

6. Stochastic Processes

Definitions and classifications

Markov processes and Markov chains

Markov chains

Matrix approach

Counting processes

Binomial process

Poisson process

Simulation of stochastic processes

Summary and conclusions

Exercises

7. Queuing Systems

Main components of a queuing system

The Little’s Law

Bernoulli single-server queuing process

Systems with limited capacity

M/M/ system

Evaluating the system’s performance

Multiserver queuing systems

Bernoulli k-server queuing process

M/M/k systems

Unlimited number of servers and M/M/∞

Simulation of queuing systems

Summary and conclusions

Exercises

III Statistics

8. Introduction to Statistics

Population and sample, parameters and statistics

Descriptive statistics

Mean

Median

Quantiles, percentiles, and quartiles

Variance and standard deviation

Standard errors of estimates

Interquartile range

Graphical statistics

Histogram

Stem-and-leaf plot

Boxplot

Scatter plots and time plots

Summary and conclusions

Exercises

9. Statistical Inference I

Parameter estimation

Method of moments

Method of maximum likelihood

Estimation of standard errors

Confidence intervals

Construction of confidence intervals: a general method

Confidence interval for the population mean

Confidence interval for the difference between two means

Selection of a sample size

Estimating means with a given precision

Unknown standard deviation

Large samples

Confidence intervals for proportions

Estimating proportions with a given precision

Small samples: Student’s t distribution

Comparison of two populations with unknown variances

Hypothesis testing

Hypothesis and alternative

Type I and Type II errors: level of significance

Level _ tests: general approach

Rejection regions and power

Standard Normal null distribution (Z-test)

Z-tests for means and proportions

Pooled sample proportion

Unknown _: T-tests

Duality: two-sided tests and two-sided confidence intervals

P-value

Variance estimator and Chi-square distribution

Confidence interval for the population variance

Testing variance

Comparison of two variances F-distribution

Confidence interval for the ratio of population variances

F-tests comparing two variances

Summary and conclusions

Exercises

10. Statistical Inference II

Chi-square tests

Testing a distribution

Testing a family of distributions

Testing independence

Nonparametric statistics

Sign test

Wilcoxon signed rank test

Mann-Whitney-Wilcoxon rank sum test

Bootstrap

Bootstrap distribution and all bootstrap samples

Computer generated bootstrap samples

Bootstrap confidence intervals

Bayesian inference

Prior and posterior

Bayesian estimation

Bayesian credible sets

Bayesian hypothesis testing

Summary and conclusions

Exercises

11. Regression

Least squares estimation

Examples

Method of least squares

Linear regression

Regression and correlation

Overfitting a model

Analysis of variance, prediction, and further inference

ANOVA and R-square

Tests and confidence intervals

Prediction

Multivariate regression

Introduction and examples

Matrix approach and least squares estimation

Analysis of variance, tests, and prediction

Model building

Extra sum of squares, partial F-tests, and variable selection

Categorical predictors and dummy variables

Summary and conclusions

Exercises

IV Appendix

12. Appendix

Data sets

Inventory of distributions

Discrete families

Continuous families

Distribution tables

Calculus review

Inverse function

Limits and continuity

Sequences and series

Derivatives, minimum, and maximum

Integrals

Matrices and linear systems