Random Sampling

Mathematical Statistics describes the mathematical underpinnings associated with the practice of statistics. The pre-requisite for this book is a calculus-based course in probability. Nearly 200 figures and dozens of Monte Carlo simulation experiments in R help develop the intuition behind the statistical methods. Real-world problems from a wide range of fields help the reader apply the statistical methods. Over 300 exercises are used to reinforce concepts and make this book appropriate for classroom use. The table of contents for this book is given below. 1. Random Sampling 2. Point Estimation 3. Interval Estimation 4. Hypothesis Testinghttps://scholarworks.wm.edu/asbookchapters/1120/thumbnail.jp

Arithmetic

Transitioning to Calculus is a comprehensive compilation of the mathematical concepts and formulas that are required of students entering their first class in calculus. The essentials of arithmetic, algebra, geometry, analytic geometry, trigonometry, and complex variables are organized into separate chapters. The purpose of this book is to provide a succinct but comprehensive list of the topics required of students entering calculus. Over 100 figures highlight the intuitive and geometric aspects of the formulas and concepts. Each chapter ends with a series of exercises (with space provided for working out a solution) that are designed to reinforce the application of the concepts and formulas. Complete solutions to the problems are included.https://scholarworks.wm.edu/asbookchapters/1117/thumbnail.jp

Statistical Modeling: Regression, Survival Analysis, and Time Series Analysis

Statistical Modeling provides an introduction to regression, survival analysis, and time series analysis for students who have completed calculus-based courses in probability and mathematical statistics. The book uses the R language to fit statistical models, conduct Monte Carlo simulation experiments and generate graphics. Over 300 exercises at the end of the chapters makes this an appropriate text for a class in statistical modeling. Part 1: RegressionChapter 1: Simple Linear Regression Chapter 2: Inference in Simple Linear Regression Chapter 3: Topics in RegressionPart II: Survival Analysis Chapter 4: Probability Models in Survival AnalysisChapter 5: Statistical Methods in Survival Analysis Chapter 6: Topics in Survival Analysis Part III: Time Series Analysis Chapter 7: Basic Methods in Time Series AnalysisChapter 8: Modeling in Time Series Analysis Chapter 9: Topics in Time Series Analysi

Introducing R

R is an open source programming language and interactive programming environment that has become the software tool of choice in data analytics. Learning Base R provides an introduction to the language for those with and without prior programming experience. It introduces the key topics that you will need to begin analyzing data and programming in R. The focus here is on the R language rather than a particular application. Within the text, there are 200 exercises to assess your R skills.https://scholarworks.wm.edu/asbookchapters/1118/thumbnail.jp

Lower Confidence Bounds for System Reliability from Binary Failure Data Using Bootstrapping

We consider the problem of determining a (1 – A) 100% lower confidence bound on the system reliability for a coherent system of k components using the failure data (yi, ni), where yi is the number of components of type i that pass the test and ni is the number of components of type i on test, i1, 2, …, k. We assume throughout that the components fail independently, e.g. no common-cause failures. The outline of the article is as follows. We begin with the case of a single (k1) component system where n components are placed on a test and y components pass the test. The Clopper-Pearson lower bound is used to provide a lower bound on the reliability. This model is then generalized to the case of multiple (k1) components. Bootstrapping is used to estimate the lower confidence bound on system reliability. We then address a weakness in the bootstrapping approach-the fact that the sample size is moot in the case of perfect test results, e.g. when yi ni for some i. This weakness is overcome by using a beta prior distribution to model the component reliability before performing the bootstrapping. Two subsections consider methods for estimating the parameters in the beta prior distribution for components with perfect test results. The first subsection considers the case when previous test results are available, and the second subsection considers the case when no previous test results are available. A simulation study compares various algorithms for calculating a lower confidence bound on the system reliability. The last section contains conclusions.https://scholarworks.wm.edu/asbookchapters/1110/thumbnail.jp

Introduction to Probability

This calculus-based introduction to probability covers all of the traditional topics, along with a secondary emphasis on Monte Carlo simulation. Examples that introduce applications from a wide range of fields help the reader apply probability theory to real-world problems. The text covers all of the topics associated with Exam P given by the Society of Actuaries. Over 100 figures highlight the intuitive and geometric aspects of probability. Over 800 exercises are used to reinforce concepts and make this text appropriate for classroom use.https://scholarworks.wm.edu/asbookchapters/1119/thumbnail.jp

Computational Algebra Applications in Reliability

Reliability analysts are typically forced to choose between using an \u27algorithmic programming language\u27 or a \u27reliability package\u27 for analyzing their models and lifetime data. This paper shows that computational languages can be used to bridge the gap to combine the flexibility of a programming language with the ease of use of a package. Computational languages facilitate the development of new statistical techniques and are excellent teaching tools. This paper considers three diverse reliability problems that are handled easily with a computational algebra language: system reliability bounds; lifetime data analysis; and model selection

Parametric Model Discrimination for Heavily Censored Survival Data

Simultaneous discrimination among various parametric lifetime models is an important step in the parametric analysis of survival data. We consider a plot of the skewness versus the coefficient of variation for the purpose of discriminating among parametric survival models. We extend the method of Cox & Oakes from complete to censored data by developing an algorithm based on a competing risks model and kernel function estimation. A by-product of this algorithm is a nonparametric survival function estimate

Symbolic ARMA Model Analysis

ARMA models provide a parsimonious and flexible mechanism for modeling the evolution of a time series. Some useful measures of these models (e.g., the autocorrelation function or the spectral density function) are tedious to compute by hand. This paper uses a computer algebra system, not simulation, to calculate measures of interest associated with ARMA models

Algorithms for Computing the Distributions of Sums of Discrete Random Variables

We present algorithms for computing the probability density function of the sum of two independent discrete random variables, along with an implementation of the algorithm in a computer algebra system. Some examples illustrate the utility of this algorithm