Matches in DBpedia 2014 for { <http://dbpedia.org/resource/McCullagh's_parametrization_of_the_Cauchy_distributions> ?p ?o. }
Showing items 1 to 25 of
25
with 100 items per page.
- McCullagh's_parametrization_of_the_Cauchy_distributions abstract "In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function isfor x real. This has median 0, and first and third quartiles respectively −1 and +1. Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one. Thus, if X has a standard Cauchy distribution and μ is any real number and σ > 0, then Y = μ + σX has a Cauchy distribution whose median is μ and whose first and third quartiles are respectively μ − σ and μ + σ.McCullagh's parametrization, introduced by Peter McCullagh, professor of statistics at the University of Chicago uses the two parameters of the non-standardised distribution to form a single complex-valued parameter, specifically, the complex number θ = μ + iσ, where i is the imaginary unit. It also extends the usual range of scale parameter to include σ < 0.Although the parameter is notionally expressed using a complex number, the density is still a density over the real line. In particular the density can be written using the real-valued parameters μ and σ, which can each take positive or negative values, aswhere the distribution is regarded as degenerate if σ = 0. An alternative form for the density can be written using the complex parameter θ = μ + iσ aswhere Differential equationTo the question "Why introduce complex numbers when only real-valued random variables are involved?", McCullagh wrote:In other words, if the random variable Y has a Cauchy distribution with complex parameter θ, then the random variable Y * defined above has a Cauchy distribution with parameter (aθ + b)/(cθ + d).McCullagh also wrote, "The distribution of the first exit point from the upper half-plane of a Brownian particle starting at θ is the Cauchy density on the real line with parameter θ." In addition, McCullagh shows that the complex-valued parameterisation allows a simple relationship to be made between the Cauchy and the "circular Cauchy distribution".".
- McCullagh's_parametrization_of_the_Cauchy_distributions wikiPageExternalLink 247.
- McCullagh's_parametrization_of_the_Cauchy_distributions wikiPageExternalLink paper18.pdf.
- McCullagh's_parametrization_of_the_Cauchy_distributions wikiPageID "2680257".
- McCullagh's_parametrization_of_the_Cauchy_distributions wikiPageRevisionID "605094830".
- McCullagh's_parametrization_of_the_Cauchy_distributions hasPhotoCollection McCullagh's_parametrization_of_the_Cauchy_distributions.
- McCullagh's_parametrization_of_the_Cauchy_distributions subject Category:Continuous_distributions.
- McCullagh's_parametrization_of_the_Cauchy_distributions subject Category:Probability_distributions.
- McCullagh's_parametrization_of_the_Cauchy_distributions type Abstraction100002137.
- McCullagh's_parametrization_of_the_Cauchy_distributions type Arrangement105726596.
- McCullagh's_parametrization_of_the_Cauchy_distributions type Cognition100023271.
- McCullagh's_parametrization_of_the_Cauchy_distributions type ContinuousDistributions.
- McCullagh's_parametrization_of_the_Cauchy_distributions type Distribution105729036.
- McCullagh's_parametrization_of_the_Cauchy_distributions type PsychologicalFeature100023100.
- McCullagh's_parametrization_of_the_Cauchy_distributions type Structure105726345.
- McCullagh's_parametrization_of_the_Cauchy_distributions comment "In probability theory, the "standard" Cauchy distribution is the probability distribution whose probability density function isfor x real. This has median 0, and first and third quartiles respectively −1 and +1. Generally, a Cauchy distribution is any probability distribution belonging to the same location-scale family as this one.".
- McCullagh's_parametrization_of_the_Cauchy_distributions label "McCullagh's parametrization of the Cauchy distributions".
- McCullagh's_parametrization_of_the_Cauchy_distributions label "Parametrización de McCullagh de las distribuciones de Cauchy".
- McCullagh's_parametrization_of_the_Cauchy_distributions sameAs Parametrización_de_McCullagh_de_las_distribuciones_de_Cauchy.
- McCullagh's_parametrization_of_the_Cauchy_distributions sameAs m.07xj4v.
- McCullagh's_parametrization_of_the_Cauchy_distributions sameAs Q6800742.
- McCullagh's_parametrization_of_the_Cauchy_distributions sameAs Q6800742.
- McCullagh's_parametrization_of_the_Cauchy_distributions sameAs McCullagh's_parametrization_of_the_Cauchy_distributions.
- McCullagh's_parametrization_of_the_Cauchy_distributions wasDerivedFrom McCullagh's_parametrization_of_the_Cauchy_distributions?oldid=605094830.
- McCullagh's_parametrization_of_the_Cauchy_distributions isPrimaryTopicOf McCullagh's_parametrization_of_the_Cauchy_distributions.