Andrew G. Glen

Computing the distribution of the product of two continuous random variables

We present an algorithm for computing the probability density function of the product of two independent random variables, along with an implementation of the algorithm in a computer algebra system. We combine this algorithm with the earlier work on transformations of random variables to create an automated algorithm for convolutions of random variables. Some examples demonstrate the algorithms application.

Computing the cumulative distribution function of the Kolmogorov-Smirnov statistic

Abstract We present an algorithm for computing the cumulative distribution function of the Kolmogorov–Smirnov test statistic D n in the all-parameters-known case. Birnbaum (1952, J. Amer. Statist. Assoc. 47, 425–441), gives an n -fold integral for the CDF of the test statistic which yields a function defined in a piecewise fashion, where each piece is a polynomial of degree n . Unfortunately, it is difficult to determine the appropriate limits of integration for computing these polynomials. Our algorithm performs the required integrations in a manner that avoids calculating the same integrals repeatedly, resulting in shorter computation time. It can be used to compute the entire CDF or just a portion of the CDF, which is more efficient for finding a critical value or a p -value associated with a hypothesis test. If the entire CDF is computed, it can be stored in memory so that various characteristics of the distribution of the test statistic (e.g., moments) can be calculated. To date, critical tables have been approximated by various techniques including asymptotic approximations, recursive formulas, and Monte Carlo simulation. Our approach yields exact critical values and significance levels. The algorithm has been implemented in a computer algebra system.

APPL: A Probability Programming Language

Statistical packages have been used for decades to analyze large datasets or to perform mathematically intractable statistical methods. These packages are not capable of working with random variables having arbitrary distributions. This article presents a prototype probability package named APPL (A Probability Programming Language) that can be used to manipulate random variables. Examples illustrate its use. A current version of the software can be obtained by contacting the third author at leemis@math.wm.edu.Statistical packages have been used for decades to analyze large datasets or to perform mathematically intractable statistical methods. These packages are not capable of working with random variables having arbitrary distributions. This article presents a prototype probability package named APPL (A Probability Programming Language) that can be used to manipulate random variables. Examples illustrate its use. A current version of the software can be obtained by contacting the third author at .

A Generalized Univariate Change-of-Variable Transformation Technique

We present a generalized version of the univariate change-of-variable technique for transforming continuous random variables. Extending a theorem from Casella and Berger [1990. Statistical Inference , Wadsworth and Brooks/Cole, Inc., Pacific Grove, CA] for many-to-1 transformations, we consider more general univariate transformations. Specifically, the transformation can range from 1-to-1 to many-to-1 on various subsets of the support of the random variable of interest. We also present an implementation of the theorem in a computer algebra system that automates the technique. Some examples demonstrate the theorems application.

THE ARCTANGENT SURVIVAL DISTRIBUTION

We present a two-parameter survival distribution that has an upside-down bathtub (UBT, or humped-shaped) hazard function. This distribution provides biostatisticians, reliability engineers, and other statisticians with a second two-parameter UBT model w..

On the Inverse Gamma as a Survival Distribution

This paper presents properties of the inverse gamma distribution and how it can be used as a survival distribution. A result is included that shows that the inverse gamma distribution always has an upside-down bathtub (UBT) shaped hazard function, thus, adding to the limited number of available distributions with this property. A review of the utility of UBT distributions is provided as well. Probabilistic properties are presented first, followed by statistical properties to demonstrate its usefulness as a survival distribution. As the inverse gamma distribution is discussed in a limited and sporadic fashion in the literature, a summary of its properties is provided in an appendix.

Experiences Teaching Probability and Statistics with Personal Laptops in the Classroom Daily

What if every day of a probability and statistics class was a computer laboratory day? How would this change your method of teaching an introductory course in probability and statistics? At the United States Military Academy at West Point, wireless laptop computers have become a permanent part of the classroom and have changed our approach to teaching. Over the last five years, we have made a concerted effort to find the improvements technology has to offer and to steer clear of the pitfalls technology can bring to the classroom. Our method of teaching a calculus-based probability and statistics course has evolved into a data-oriented approach to understanding distributions. We present some methods we have developed that use spreadsheets and mathematical software systems to create an environment that helps students understand the foundations of probability theory and statistical inference.

Applying Bootstrap Methods to System Reliability

We present a fully enumerated bootstrap method to find the empirical system lifetime distribution for a coherent system modeled by a reliability block diagram. Given failure data for individual components of a coherent system, the bootstrap empirical system lifetime distribution derived here will be free of resampling error. We further derive distribution-free expressions for the bias associated with the bootstrap method for estimating the mean system lifetimes of parallel and series systems with statistically identical components. We show that bootstrapping underestimates the mean system lifetime for parallel systems and overestimates the mean system lifetime for series systems, although both bootstrap estimates are asymptotically unbiased. The expressions for the bias are evaluated for several popular parametric lifetime distributions. Supplementary materials for this article are available online.

Transformations of Random Variables

This chapter presents a generalized version of the univariate change-of-variable technique for transforming continuous random variables. Extending a theorem from Casella and Berger [16] for many–to–1 transformations, we consider more general univariate transformations. Specifically, the transformation can range from 1–to–1 to many–to–1 on various subsets of the support of the random variable of interest. We also present an implementation of the theorem in APPL and present four examples.

Survival distributions based on the incomplete gamma function ratio

A method to produce new families of probability distributions is presented based on the incomplete gamma function ratio. The distributions distributions produced also can include a number of popular univariate survival distributions, including the gamma, chi-square, exponential, and half-normal. Examples that demonstrate the generation of new distributions are provided.