Mathew Penrose
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0001
- Subject:
- Mathematics, Probability / Statistics
This introductory chapter contains a general discussion of both the historical and the applied background behind the study of random geometric graphs. A brief overview is presented, along with some ...
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This introductory chapter contains a general discussion of both the historical and the applied background behind the study of random geometric graphs. A brief overview is presented, along with some standard definitions in graph theory and probability theory. Specific terminology is introduced for two limiting regimes in the choice of r=r(n) (namely the thermodynamic limit where the mean vertex degree is made to approach a finite constant) and the connectivity regime (where it grows logarithmically with n). Some elementary probabilistic results are given on large deviations for the binomial and Poisson distribution, and on Poisson point processes.Less
This introductory chapter contains a general discussion of both the historical and the applied background behind the study of random geometric graphs. A brief overview is presented, along with some standard definitions in graph theory and probability theory. Specific terminology is introduced for two limiting regimes in the choice of r=r(n) (namely the thermodynamic limit where the mean vertex degree is made to approach a finite constant) and the connectivity regime (where it grows logarithmically with n). Some elementary probabilistic results are given on large deviations for the binomial and Poisson distribution, and on Poisson point processes.
Stephen J. Blundell and Katherine M. Blundell
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199562091
- eISBN:
- 9780191718236
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562091.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter defines some basic concepts in probability theory. It begins by stating that the probability of occurrence of a particular event, taken from a finite set of possible events, is zero if ...
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This chapter defines some basic concepts in probability theory. It begins by stating that the probability of occurrence of a particular event, taken from a finite set of possible events, is zero if that event is impossible, is one if that event is certain, and takes a value somewhere in between zero and one if that event is possible but not certain. It considers two different types of probability distribution: discrete and continuous. Variance, linear transformation, independent variables, and binomial distribution are also discussed.Less
This chapter defines some basic concepts in probability theory. It begins by stating that the probability of occurrence of a particular event, taken from a finite set of possible events, is zero if that event is impossible, is one if that event is certain, and takes a value somewhere in between zero and one if that event is possible but not certain. It considers two different types of probability distribution: discrete and continuous. Variance, linear transformation, independent variables, and binomial distribution are also discussed.
Therese M. Donovan and Ruth M. Mickey
- Published in print:
- 2019
- Published Online:
- July 2019
- ISBN:
- 9780198841296
- eISBN:
- 9780191876820
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198841296.003.0008
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies
This chapter focuses on probability mass functions. One of the primary uses of Bayesian inference is to estimate parameters. To do so, it is necessary to first build a good understanding of ...
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This chapter focuses on probability mass functions. One of the primary uses of Bayesian inference is to estimate parameters. To do so, it is necessary to first build a good understanding of probability distributions. This chapter introduces the idea of a random variable and presents general concepts associated with probability distributions for discrete random variables. It starts off by discussing the concept of a function and goes on to describe how a random variable is a type of function. The binomial distribution and the Bernoulli distribution are then used as examples of the probability mass functions (pmf’s). The pmfs can be used to specify prior distributions, likelihoods, likelihood profiles and/or posterior distributions in Bayesian inference.Less
This chapter focuses on probability mass functions. One of the primary uses of Bayesian inference is to estimate parameters. To do so, it is necessary to first build a good understanding of probability distributions. This chapter introduces the idea of a random variable and presents general concepts associated with probability distributions for discrete random variables. It starts off by discussing the concept of a function and goes on to describe how a random variable is a type of function. The binomial distribution and the Bernoulli distribution are then used as examples of the probability mass functions (pmf’s). The pmfs can be used to specify prior distributions, likelihoods, likelihood profiles and/or posterior distributions in Bayesian inference.
James Davidson
- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0008
- Subject:
- Economics and Finance, Econometrics
Specializing the concepts of Ch. 7 to the case of real variables, this chapter introduces distribution functions, discrete and continuous distributions, and describes examples such as the binomial, ...
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Specializing the concepts of Ch. 7 to the case of real variables, this chapter introduces distribution functions, discrete and continuous distributions, and describes examples such as the binomial, uniform, Gaussian, and Cauchy distributions. It then treats multivariate distributions and the concept of independence.Less
Specializing the concepts of Ch. 7 to the case of real variables, this chapter introduces distribution functions, discrete and continuous distributions, and describes examples such as the binomial, uniform, Gaussian, and Cauchy distributions. It then treats multivariate distributions and the concept of independence.
Andy Hector
- Published in print:
- 2015
- Published Online:
- March 2015
- ISBN:
- 9780198729051
- eISBN:
- 9780191795855
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198729051.003.0009
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies, Ecology
This chapter looks at three of the main types of generalized linear model (GLM). GLMs using the Poisson distribution are a good starting place when dealing with integer count data. The default log ...
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This chapter looks at three of the main types of generalized linear model (GLM). GLMs using the Poisson distribution are a good starting place when dealing with integer count data. The default log link function prevents the prediction of negative counts and the Poisson distribution models the variance (approximately equal to the mean). GLMs with a binomial distribution are designed for the analysis of binomial counts (how many times something occurred relative to the total number of possible times it could have occurred). A logistic link function constrains predictions to be above zero and below the maximum using the S-shaped logistic curve. Overdispersion can be diagnosed and dealt with using a quasi-maximum likelihood extension to GLM analysis. Binomial GLMs can also be used to analyse binary data as a special case with some minor differences to the analysis introduced by the constrained nature of the binary data.Less
This chapter looks at three of the main types of generalized linear model (GLM). GLMs using the Poisson distribution are a good starting place when dealing with integer count data. The default log link function prevents the prediction of negative counts and the Poisson distribution models the variance (approximately equal to the mean). GLMs with a binomial distribution are designed for the analysis of binomial counts (how many times something occurred relative to the total number of possible times it could have occurred). A logistic link function constrains predictions to be above zero and below the maximum using the S-shaped logistic curve. Overdispersion can be diagnosed and dealt with using a quasi-maximum likelihood extension to GLM analysis. Binomial GLMs can also be used to analyse binary data as a special case with some minor differences to the analysis introduced by the constrained nature of the binary data.
Robert N. Compton and Michael A. Duncan
- Published in print:
- 2015
- Published Online:
- December 2015
- ISBN:
- 9780198742975
- eISBN:
- 9780191816932
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198742975.003.0019
- Subject:
- Physics, Atomic, Laser, and Optical Physics
Laser Raman spectroscopy has been a powerful tool for the characterization of molecules for many years. This chapter describes a novel technique for the recording of Raman spectra under liquid ...
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Laser Raman spectroscopy has been a powerful tool for the characterization of molecules for many years. This chapter describes a novel technique for the recording of Raman spectra under liquid nitrogen (RUN) which reduces sample burning and increases resolution. An introduction to the theory of Raman spectroscopy is followed by a description of how to perform RUN spectroscopy with examples using CCl4, CS2, and ferrocene. Computations using the Gaussian program are also described for the molecules studied. The calculation of isotopic distributions is also treated and applied to the interpretation of the spectra.Less
Laser Raman spectroscopy has been a powerful tool for the characterization of molecules for many years. This chapter describes a novel technique for the recording of Raman spectra under liquid nitrogen (RUN) which reduces sample burning and increases resolution. An introduction to the theory of Raman spectroscopy is followed by a description of how to perform RUN spectroscopy with examples using CCl4, CS2, and ferrocene. Computations using the Gaussian program are also described for the molecules studied. The calculation of isotopic distributions is also treated and applied to the interpretation of the spectra.
