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Campbell's Theorem (probability)
In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem,D. Stoyan, W. S. Kendall, J. Mecke. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995. also known as Campbell's formula, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving moment measures and factorial moment measures that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the theory of point processes and queueing theory as well as the related fields stochastic geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Statistics
Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early development, using different analytic approaches and applied in fields as diverse as astronomy, with its studies of the placement of galaxies in the cosmos, to chip fabrication engineering, with its use of "place and route" algorithms to build complex wiring structures. In a more restricted sense, spatial analysis is the technique applied to structures at the human scale, most notably in the analysis of geographic data or transcriptomics data. Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved, but form the basis for current research. The most fundamental of these is the problem of defining the spatial location of the entities being studied. Classification of the techniques of spatial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment-generating Function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions. As its name implies, the moment-generating function can be used to compute a distribution’s moments: the ''n''th moment about 0 is the ''n''th derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases. The moment-generating func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harry Bateman
Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician with a specialty in differential equations of mathematical physics. With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poincare to a more expansive conformal group of spacetime leaving Maxwell's equations invariant. Moving to the U.S.A., and obtaining a Ph.D. in geometry with Frank Morley, he became a professor of mathematics at California Institute of Technology. There he taught fluid dynamics to students going into aerodynamics with Theodore von Karman. Bateman made a broad survey of applied differential equations in his Gibbs Lecture in 1943 titled "The control of an elastic fluid". Biography Bateman was born in Manchester, England on 29 May 1882. He first grew to love mathematics at Manchester Grammar School, and, in his final year, won a scholarship to Trinity College, Cambridge. Bateman studied with coach Robert Alfred Herman preparing for Cambridge Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alpha Particle
Alpha particles, also called alpha rays or alpha radiation, consist of two protons and two neutrons bound together into a particle identical to a helium-4 nucleus. They are generally produced in the process of alpha decay, but may also be produced in other ways. Alpha particles are named after the first letter in the Greek alphabet, α. The symbol for the alpha particle is α or α2+. Because they are identical to helium nuclei, they are also sometimes written as or indicating a helium ion with a +2 charge (missing its two electrons). Once the ion gains electrons from its environment, the alpha particle becomes a normal (electrically neutral) helium atom . Alpha particles have a net spin of zero. Due to the mechanism of their production in standard alpha radioactive decay, alpha particles generally have a kinetic energy of about 5 MeV, and a velocity in the vicinity of 4% of the speed of light. (See discussion below for the limits of these figures in alpha decay.) They are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Geiger
Johannes Wilhelm "Hans" Geiger (; ; 30 September 1882 – 24 September 1945) was a German physicist. He is best known as the co-inventor of the detector component of the Geiger counter and for the Geiger–Marsden experiment which discovered the atomic nucleus. He was the brother of meteorologist and climatologist Rudolf Geiger. Biography Geiger was born at Neustadt an der Haardt, Germany. He was one of five children born to the Indologist Wilhelm Ludwig Geiger, who was a professor at the University of Erlangen. In 1902, Geiger started studying physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ... and mathematics at the University of Erlangen and was awarded a doctorate in 1906. His thesis was on electrical discharges through gases. He received a fellowship to the Victo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernest Rutherford
Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest experimentalist since Michael Faraday (1791–1867). Apart from his work in his homeland, he spent a substantial amount of his career abroad, in both Canada and the United Kingdom. In early work, Rutherford discovered the concept of radioactive half-life, the radioactive element radon, and differentiated and named alpha and beta radiation. This work was performed at McGill University in Montreal, Quebec, Canada. It is the basis for the Nobel Prize in Chemistry he was awarded in 1908 "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances", for which he was the first Oceanian Nobel laureate, and the first to perform the awarded work in Canada. In 1904, he was elected as a member to the Am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Tubes
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied. The type known as a thermionic tube or thermionic valve utilizes thermionic emission of electrons from a hot cathode for fundamental electronic functions such as signal amplification and current rectification. Non-thermionic types such as a vacuum phototube, however, achieve electron emission through the photoelectric effect, and are used for such purposes as the detection of light intensities. In both types, the electrons are accelerated from the cathode to the anode by the electric field in the tube. The simplest vacuum tube, the diode (i.e. Fleming valve), invented in 1904 by John Ambrose Fleming, contains only a heated electron-emitting cathode and an anode. Electrons can only flow in one direction through the device—from the cathode to the anode. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shot Noise
Shot noise or Poisson noise is a type of noise which can be modeled by a Poisson process. In electronics shot noise originates from the discrete nature of electric charge. Shot noise also occurs in photon counting in optical devices, where shot noise is associated with the particle nature of light. Origin In a statistical experiment such as tossing a fair coin and counting the occurrences of heads and tails, the numbers of heads and tails after many throws will differ by only a tiny percentage, while after only a few throws outcomes with a significant excess of heads over tails or vice versa are common; if an experiment with a few throws is repeated over and over, the outcomes will fluctuate a lot. From the law of large numbers, one can show that the relative fluctuations reduce as the reciprocal square root of the number of throws, a result valid for all statistical fluctuations, including shot noise. Shot noise exists because phenomena such as light and electric current co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermionic Emission
Thermionic emission is the liberation of electrons from an electrode by virtue of its temperature (releasing of energy supplied by heat). This occurs because the thermal energy given to the charge carrier overcomes the work function of the material. The charge carriers can be electrons or ions, and in older literature are sometimes referred to as thermions. After emission, a charge that is equal in magnitude and opposite in sign to the total charge emitted is initially left behind in the emitting region. But if the emitter is connected to a battery, the charge left behind is neutralized by charge supplied by the battery as the emitted charge carriers move away from the emitter, and finally the emitter will be in the same state as it was before emission. The classical example of thermionic emission is that of electrons from a hot cathode into a vacuum (also known as thermal electron emission or the Edison effect) in a vacuum tube. The hot cathode can be a metal filament, a coated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Robert Campbell
Norman Robert Campbell (1880–1949) was an English physicist and philosopher of science. Early life Norman Robert Campbell was born in 1880. He was the son of William Middleton Campbell, Governor of the Bank of England, and his wife Edith Agneta Bevan. He was educated at Eton College and at Trinity College, Cambridge, graduating B.A. in 1902. Career Campbell became a fellow at Trinity College, Cambridge in 1902. He was also a research assistant at the Cavendish Laboratory under the direction of J. J. Thomson. He became an honorary fellow in physics research at Leeds University in 1913. During World War I Campbell was at the National Physical Laboratory, working under Clifford Paterson. They investigated spark discharge in internal combustion engines. He then worked from 1919 to 1944 as a member of the research staff at the Hirst Research Centre of the General Electric Company England, set up in Wembley Middlesex in 1919. While at Hirst Research he encouraged unqualified st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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