Bivariate Probability Distributions and Sampling Distributions

2016 ◽  
pp. 234-287
Author(s):  
William M. Mendenhall ◽  
Terry L. Sincich
Keyword(s):  
Probability Distributions ◽  
Sampling Distributions ◽  
Bivariate Probability

scholarly journals Objective determination of the sea age of Atlantic salmon from the sizes and dates of capture of individual fish

2010 ◽  
Vol 68 (1) ◽  
pp. 130-143 ◽  
Author(s):  
Philip J. Bacon ◽  
William S. C. Gurney ◽  
Eddie McKenzie ◽  
Bryce Whyte ◽  
Ronald Campbell ◽  
...  
Keyword(s):  
Atlantic Salmon ◽  
Fishery Management ◽  
East Coast ◽  
Probability Distributions ◽  
Age Classes ◽  
Individual Fish ◽  
Bivariate Probability ◽  
Management And Conservation ◽  
Objective Determination

Abstract Bacon, P. J., Gurney, W. S. C., McKenzie, E., Whyte, B., Campbell, R., Laughton, R., Smith, G., and MacLean, J. 2011. Objective determination of the sea age of Atlantic salmon from the sizes and dates of capture of individual fish. – ICES Journal of Marine Science, 68: 130–143. The sea ages of Atlantic salmon indicate crucial differences between oceanic feeding zones that have important implications for conservation and management. Historical fishery-catch records go back more than 100 years, but the reliability with which they discriminate between sea-age classes is uncertain. Research data from some 188 000 scale-aged Scottish salmon that included size (length, weight) and seasonal date of capture on return to the coast were investigated to devise a means of assigning sea age to individual fish objectively. Two simple bivariate probability distributions are described that discriminate between 1SW and 2SW fish with 97% reliability, and between 2SW and 3SW fish with 70% confidence. The same two probability distributions achieve this accuracy across five major east coast Scottish rivers and five decades. They also achieve the same exactitude for a smaller recent dataset from the Scottish west coast, from the River Tweed a century ago (1894/1895), and for salmon caught by rod near the estuary. More surprisingly, they also achieve the same success for rod-caught salmon taken at beats remote from the estuary and including capture dates when some fish could have been in the river for a few months. The implications of these findings for fishery management and conservation are discussed.

scholarly journals Understanding the Central Limit Theorem the Easy Way: A Simulation Experiment

Proceedings ◽  
2018 ◽  
Vol 2 (21) ◽  
pp. 1322
Author(s):  
Monica E. Brussolo
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Electronic Device ◽  
Probability Distributions ◽  
Online Simulation ◽  
The Central Limit Theorem ◽  
Sampling Distributions ◽  
Initial Stage ◽  
Random Samples

Using a simulation approach, and with collaboration among peers, this paper is intended to improve the understanding of sampling distributions (SD) and the Central Limit Theorem (CLT) as the main concepts behind inferential statistics. By demonstrating with a hands-on approach how a simulated sampling distribution performs when the data used has different probability distributions, we expect to clarify the notion of the Central Limit Theorem, and the use of samples in the hypothesis testing process for populations. This paper will discuss an initial stage to create random samples from a given population (using Excel) with collaboration of the students, which has been tested in the classroom. Then, based on that experience, a second stage in which we created an online simulation, controlled by the professor, and in which the students will participate during class time using an electronic device connected to internet. Students will create simple random samples from a variety of probability distributions simulated online in a collaborative way. Once the samples are generated, the instructor will combine and summarize the resulting sample statistics using histograms and the results will be discussed with the students. The objective is to teach some of the central topics of introductory statistics, the Central Limit Theorem and sampling distributions with an interactive and engaging approach.

scholarly journals The tenets of indirect inference in Bayesian models

2021 ◽  
Author(s):  
Dmytro Perepolkin ◽  
Benjamin Goodrich ◽  
Ullrika Sahlin
Keyword(s):  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Bayesian Models ◽  
Quantile Function ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Indirect Inference ◽  
Root Finding ◽  
Sampling Distributions ◽  
Definition Of

This paper extends the application of Bayesian inference to probability distributions defined in terms of its quantile function. We describe the method of *indirect likelihood* to be used in the Bayesian models with sampling distributions which lack an explicit cumulative distribution function. We provide examples and demonstrate the equivalence of the "quantile-based" (indirect) likelihood to the conventional "density-defined" (direct) likelihood. We consider practical aspects of the numerical inversion of quantile function by root-finding required by the indirect likelihood method. In particular, we consider a problem of ensuring the validity of an arbitrary quantile function with the help of Chebyshev polynomials and provide useful tips and implementation of these algorithms in Stan and R. We also extend the same method to propose the definition of an *indirect prior* and discuss the situations where it can be useful

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