Mathematical forms of the distribution of the product of two normal variables

This paper finds the mathematical forms of the distribution of the product where x and x follow a bivariate normal distribution In this paper the distribution when PT0 is expressed as an integral, a new, fundamental result. From this general form, six different cases can be distinguished depending on what is known about the parameters and p. The special cases are Aroian $year:1959 and (6) Additionally, we prove that if and as the distribution of the product approaches the Type III distribution. When p=0# Aroian $year:1959 and Aroian and Meeker $year:1977, give tables for various values of 6., 6 . The results in this paper will be used to provide brief tables for p^O in a separate paper     

Revisions of the Fish Invasiveness Screening Kit (FISK) for its Application in Warmer Climatic Zones,with Particular Reference to Peninsular Florida

The initial version (v1) of the Fish Invasiveness Scoring Kit (FISK) was adapted from the Weed Risk Assessment of Pheloung, Williams, and Halloy to assess the potential invasiveness of nonnative freshwater fishes in the United Kingdom. Published applications of FISK v1 have been primarily in temperate‐zone countries (Belgium, Belarus, and Japan), so the specificity of this screening tool to that climatic zone was not noted until attempts were made to apply it in peninsular Florida. To remedy this shortcoming, the questions and guidance notes of FISK v1 were reviewed and revised to improve clarity and extend its applicability to broader climatic regions, resulting in changes to 36 of the 49 questions. In addition, upgrades were made to the software architecture of FISK to improve overall computational speed as well as graphical user interface flexibility and friendliness. We demonstrate the process of screening a fish species using FISK v2 in a realistic management scenario by assessing the Barcoo grunter Scortum barcoo (Terapontidae), a species whose management concerns are related to its potential use for aquaponics in Florida. The FISK v2 screening of Barcoo grunter placed the species into the lower range of medium risk (score = 5), suggesting it is a permissible species for use in Florida under current nonnative species regulations. Screening of the Barcoo grunter illustrates the usefulness of FISK v2 as a proactive tool serving to inform risk management decisions, but the low level of confidence associated with the assessment highlighted a dearth of critical information on this species.     

Lattice path combinatorics for multiple product identities

Lattice paths are enumerated as walks on a lattice under the Delannoy criterion of vertical, horizontal and upward diagonal steps. Delannoy recursions are generalized for arbitrary weights in each of these directions, where the row elements of the Delannoy triangles correspond to lattice paths for all walks at a given length.     

Regional Religions?: Extending the “Semi-involuntary” Thesis of African-American Religious Participation

This research examines religious affiliation and church attendance among African-Americans in three different regions of the United States to evaluate the thesis that the Black church represents a semi-involuntary institution shaped by historical dynamics of segregation in the rural South. We extend the analyses of others who have found the rural South to have distinctive church participation patterns by examining two nationally representative data sets (the 1972–1996 General Social Surveys and the 1984 National Alcohol Study). We explore both level and type of church attendance of African-Americans, and how patterns differ by region. Further, we refine prior analyses by (1) differentiating between members of historically White and conservative churches from those in the black mainline, (2) examining racial segregation, and (3) focusing on the type of church attendance (rather than just overall level). Ordinary Least Squares (OLS) and logistic regression models support the thesis that the rural South exhibits some distinctive patterns that make the semi-involuntary institution notion a useful concept, although patterns not predicted by the thesis are also found. The semi-involuntary thesis is also used to illuminate some church attendance patterns observed outside the rural South.     

Some aspects of the reparametrization of statistical models

Definitions are given for orthogonal parameters in the context of Bayesian inference and likelihood inference. The exact orthogonalizing transformations are derived for both cases, and the connection between the two settings is made precise. These parametrizations simplify the interpretation of likelihood functions and posterior distributions. Further, they make numerical maximization and integration procedures easier to apply. Several applications are studied.     

DYSFUNCTIONAL MARITAL CONFLICT: AN INTEGRATIVE INTERPERSONAL-INTRAPSYCHIC MODEL*

An integrative interpersonal-intrapsychic model of dysfunctional marital conflict is presented and research support for the model is reviewed. The therapeutic applications of the model are discussed and illustrated with a clinical example.