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Topic: Infinitely divisible probability distribution

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Niklas Luhmann

	PlanetMath: infinitely divisible random variable
		A distribution function is said to be infinitely divisible if it is the distribution function of an infinitely divisible random variable.
		Some examples of infinitely divisible distribution functions, besides those that are stable, are the gamma distributions, negative binomial distributions, and compound Poisson distributions.
		This is version 4 of infinitely divisible random variable, born on 2006-11-24, modified 2006-11-24.
	www.planetmath.org /encyclopedia/InfinitelyDivisible.html (152 words)

	Infinite divisibility - Wikipedia, the free encyclopedia
		By contrast, the ring of integers is not infinitely divisible.
		The Poisson distributions, the normal distributions, and the gamma distributions are infinitely divisible probability distributions.
		This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.
	en.wikipedia.org /wiki/Infinite_divisibility (1394 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		There he shows how infinite divisibility involves the idea that there is some extended item, such as an apple, which can be divided infinitely many times, where one never divides down to point, or to atoms of any sort.
		The Poisson distribution, the negative binomial distribution, and the Gamma distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution and all other members of the stable distribution family.
		This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Infinite_divisibility (1324 words)

	Poisson distribution
		The binomial distribution with parameters n and λ/n, i.e., the probability distribution of the number of successes in n trials, with probability λ/n of success on each trial, approaches the Poisson distribution with expected value λ as n approaches infinity.
		The higher moments of the Poisson distribution are Touchard polynomials in λ, whose coefficients have a combinatorial meaning.
		Accordingly, the Poisson distribution is sometimes called the law of small numbers because it is the probability distribution of the number of occurrences of an event that happens rarely but has very many opportunities to happen.
	publicliterature.org /en/wikipedia/p/po/poisson_distribution.html (821 words)

	Normal Distribution Encyclopedia Article @ Ordinarily.org (Site not responding. Last check: 2007-10-24)
		The fundamental importance of the normal distribution as a model of quantitative phenomena in the natural and behavioral sciences is due to the central limit theorem (the proof of which requires rather advanced undergraduate mathematics).
		The normal distribution also arises in many areas of statistics: for example, the sampling distribution of the mean is approximately normal, even if the distribution of the population the sample is taken from is not normal.
		The normal distribution was first introduced by Abraham de Moivre in an article in 1734 (reprinted in the second edition of his The Doctrine of Chances, 1738) in the context of approximating certain binomial distributions for large n.
	www.ordinarily.org /encyclopedia/Normal_distribution (4069 words)

	Infinite divisibility - Definition, explanation
		In physics, the question of whether matter is infinitely divisible is the question of whether it is true that no matter how small the pieces into which a physical object has been cut, they can be split further.
		Physical space has often been regarded as infinitely divisible: it was thought that any region in space, no matter how small, could be further split.
		The Poisson distributions, the normal distributions, and the gamma distributions are infinitely divisible probability distributions.
	www.calsky.com /lexikon/en/txt/i/in/infinite_divisibility.php (726 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-24)
		Probability distributions in function spaces are usually required to satisfy some regularity property, usually formulated as separability but also admitting a characterization in different terms (see Separable process and also [3]).
		Many of the probability distributions that appear in the specific problems in probability theory and mathematical statistics have been known for a long time and are connected with the basic probability schemes [4].
		A uniform distribution, usually considered as a mathematical way of expressing that outcomes of an experiment are equally possible, can also be obtained as a limit distribution (say, by considering sums of large numbers of random variables or some other random variables with sufficiently smooth and "spread out" distributions modulo 1).
	eom.springer.de /P/p074900.htm (687 words)

	Normal distribution Summary
		The fundamental importance of the normal distribution as model of quantitative phenomena in the natural and behavioral sciences, is due to the central limit theorem (the proof of which requires rather advanced undergraduate mathematics).
		The normal distribution also arises in many areas of statistics: for example, the sampling distribution of the mean is approximately normal, even if the distribution of the population the sample is taken from is not normal.
		The normal distribution was first introduced by Abraham de Moivre in an article in 1734 (reprinted in the second edition of his The Doctrine of Chances, 1738) in the context of approximating certain binomial distributions for large n.
	www.bookrags.com /Normal_distribution (4551 words)

	Reference.com/Encyclopedia/Infinite divisibility
		There he shows how infinite divisibility involves the idea that there is some extended item, such as an apple, which can be divided infinitely many times, where one never divides down to point, or to atoms of any sort.
		The Poisson distribution and the Gamma distribution are both examples of infinitely divisible distributions as are the Normal distribution, Cauchy distribution and all other members of the stable distribution family.
		This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.
	www.reference.com /browse/wiki/Infinite_divisibility (1330 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-24)
		The composition of infinitely-divisible distributions and the limit of weakly-convergent sequences of infinitely-divisible distributions are again infinitely divisible.
		; 2) the probability distribution of the increment
		The importance of the role played in the limit theorems of probability theory by infinitely-divisible distributions is due to the fact that these and only these distributions can be the limit distributions for sums of independent random variables subject to the requirement of asymptotic negligibility.
	eom.springer.de /I/i050910.htm (698 words)

	Bambooweb: Infinitely divisible probability distribution
		In physics, the question of whether matter is infinitely divisible is the question of whether it is true that no matter how small the pieces into which a phyiscal object has been cut, they can be split further.
		Similarly, time is infinitely divisible if any interval of time can be further split; the alternative is that time comes in discrete moments.
		To say that the field of rational numbers is infinitely divisible means that between any two rational numbers there is another rational number.
	www.bambooweb.com /articles/i/n/Infinitely_divisible_probability_distribution.html (534 words)

	The Probability Distributions of Rainstorm Intensities: Levy Stable Distributions?
		There were contributing factors in the way the dams were managed but the root cause was probably the underdesign of the dams for the extreme storms of the area and this most likely came from the flaws in the probability model used to estimate the probabilities of different size storms.
		To use this result to estimate the probabilities of extreme cases without any basis for the belief that the probability distribution is, in fact, a negative exponential curve is very shaky.
		For non-normal distribution ν has a finite non-negative value but it is not the same as the standard deviation, which for non-normal stable distributions is infinite.
	www2.sjsu.edu /faculty/watkins/storms.htm (2155 words)

	Infinitely Divisible Random Variables and Their Characteristic Functions
		One key element of that structure is the theory of infinitely divisible random variables.
		A variable z is said to be infinitely divisible if all values of n from 1 up it can be represented as the sum of n independent, identically distributed randome variables.
		For this variable to be infinitely divisible it would at least have to be divisible; i.e., representable as the sum of two independent, identically distributed random variables.
	www.sjsu.edu /faculty/watkins/infdiv.htm (741 words)

	Poisson distribution Summary
		In probability theory and statistics, the Poisson distribution is a discrete probability distribution.
		For temporally distributed events, the Poisson distribution is the probability distribution of the number of events that would occur within a preset time, the Erlang distribution is the probability distribution of the amount of time until the nth event.
		Skellam distribution, the distribution of the difference of two Poisson variates, not necessarily from the same parent distribution.
	www.bookrags.com /Poisson_distribution (2272 words)

	Compound Poisson distribution - Wikipedia, the free encyclopedia
		In probability theory, a compound Poisson distribution is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables.
		are identically distributed random variables that are mutually independent and also independent of N.
		It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions.
	en.wikipedia.org /wiki/Compound_Poisson_distribution (259 words)

	Reference.com/Encyclopedia/Cauchy distribution
		As a probability distribution, it is known as the Cauchy distribution while among physicists it is known as the Lorentz distribution or the Breit-Wigner distribution.
		The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined.
		Thus the Cauchy distribution is a ratio distribution.
	www.reference.com /browse/wiki/Cauchy_distribution (777 words)

	Probability & Statistics 3
		In probability theory, to say that two events are independent intuitively means that knowing whether or not one of them occurs makes it neither more probable nor less probable that the other occurs.
		The normal distribution is a convenient model of quantitative phenomena in the natural and behavioral sciences.
		The normal distribution was first introduced by de Moivre in an article in 1733 (reprinted in the second edition of his The Doctrine of Chances, 1738) in the context of approximating certain binomial distributions for large n.
	www.physicsarchives.com /statistics3.htm (2349 words)

	Prior probability - Psychology Wiki (Site not responding. Last check: 2007-10-24)
		A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence.
		The posterior probability is computed from the prior and the likelihood function via Bayes' theorem.
		We could specify, say, a normal distribution as the prior for his speed, but alternatively we could specify a normal prior for the time he takes to complete 100 metres, which is proportional to the reciprocal of the first prior.
	psychology.wikia.com /wiki/Prior_probability (1530 words)

	Interpretations of Probability (Stanford Encyclopedia of Philosophy)
		Entropy is a measure of the lack of ‘informativeness’ of a probability distribution.
		Probability is thought of as a physical propensity, or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome.
		The laws of probability then are claimed to be constraints on these estimates: putative necessary conditions for minimizing her ‘losses’ in a broad sense, be they monetary, or measured by distances from the assignments of these experts.
	plato.stanford.edu /entries/probability-interpret (15166 words)

	PlanetMath: Lévy process
		In probability theory, a Lévy process, named after the French mathematician Paul Pierre Lévy is any continuous-time stochastic process that starts at 0, admits càdlàg (right-continuous with left limits) modification and has “stationary independent increments”.
		The probability distributions of the increments of any Lévy process are infinitely divisible.
		There is a Lévy process for each infinitely divisible probability distribution.
	planetmath.org /encyclopedia/LevyProcess.html (350 words)

	[No title]
		In this text the problem about possible limit distributions of normalized sums of independent random variables or more generally about limit distribution of sums of random variables in a row from a triangular array (whose elements in the same row are independent) is discussed.
		The second part contains the necessary and sufficient condition for the existence of a limit distribution for sums of random variables in fixed rows of a triangular array of random variables if the elements in a row of this triangular array are independent, and they satisfy the so-called uniform smallness condition.
		The necessary and sufficient condition for the existence of a limit distribution for the sums of random variables in fixed rows of a triangular array can be expressed by means of certain canonical measures on the real line which are simple transforms of the distribution functions of the random variables we consider.
	www.renyi.hu /~major/probability/divisible.html (719 words)

	Infinite divisibility (Site not responding. Last check: 2007-10-24)
		In physics, the question of whether matter is infinitely divisible is the question of whether it is true that no matter how small the pieces into which a physical object has been cut, they can be split further.
		Later, those objects to which the name atom had been assigned were found to be further divisible, but the word atom nonetheless continues to refer to them.
		Thus time in market records is not infinitely divisible.
	infinite-divisibility.iqnaut.net (693 words)

	Math Forum Discussions
		infinitely divisible and lots of others are not.
		distribution of each of which is G, then the distribution of their
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/message.jspa?messageID=3742766&tstart=0 (342 words)

	Math Forum - Ask Dr. Math Archives: Middle School Probability
		My co-worker prefers to bet the same set of five lottery numbers every time, but I say that the probability is the same if you randomly select any set five numbers for the same period of time.
		I know the probability of getting a D is therefore 1 out of 2, and E is 1 out of 6.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/drmath/sets/mid_probability.html (899 words)

	STATSnetBASE: Statistical Sciences Online (Site not responding. Last check: 2007-10-24)
		The theory of infinitely divisible distributions plays a fundamental role in several parts of theoretical probability, such as the central limit problem and the theory of processes with stationary independent increments or L´evy processes, which lie at the roots of our subject.
		In models that require random variables to be the sum of several independent quantities with the same distribution, a convenient assumption is infinite divisibility of these random variables; this situation occurs in biology and insurance.
		Theoretically, the problem of identifying the infinitely divisible distributions is completely solved by the canonical representations of their characteristic functions.
	www.statsnetbase.com /ejournals/books/book_summary/summary.asp?id=1361 (240 words)

	Distribution Drop (Site not responding. Last check: 2007-10-24)
		Disdrometer - A disdrometer is an instrument used to measure the drop size distribution and velocity of falling hydrometeors.
		Cantor distribution - The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.
		This distribution is not absolutely continuous with respect to Lebesgue measure, so it has no probability density function; neither is it discrete, since it has no point-masses; nor is it even a mixture of a discrete probability distribution with one that has a density function.
	www.talctrain.com /Distribution-Drop.jsp (1730 words)

	Random walks, Brownian motion, Statistics and Probability
		To be divisible by p and q is equivalent to being divisible by pq and consequently the density of the new set is 1/pq.
		For twins the "decay constant'' decreases as the reciprocal of the logarithm of the length of the sequence, whereas for triplets the falloff is faster: decreasing as the square of the reciprocal of the logarithm of the number of primes.
		Endo and A. Khrennikov, "On the annihilators of the p-adic Gaussian distributions", Comm.
	secamlocal.ex.ac.uk /~mwatkins/zeta/physics6.htm (6581 words)

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