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13,500 | Many cultures have stories describing the origin of the world and universe. Cultures generally regard these stories as having some truth. There are however many differing beliefs in how these stories apply amongst those believing in a supernatural origin, ranging from a god directly creating the universe as it is now to a god just setting the "wheels in motion" (for example via mechanisms such as the big bang and evolution). | https://en.wikipedia.org/wiki?curid=31880 |
13,501 | Ethnologists and anthropologists who study myths have developed various classification schemes for the various themes that appear in creation stories. For example, in one type of story, the world is born from a world egg; such stories include the Finnish epic poem "Kalevala", the Chinese story of Pangu or the Indian Brahmanda Purana. In related stories, the universe is created by a single entity emanating or producing something by him- or herself, as in the Tibetan Buddhism concept of Adi-Buddha, the ancient Greek story of Gaia (Mother Earth), the Aztec goddess Coatlicue myth, the ancient Egyptian god Atum story, and the Judeo-Christian Genesis creation narrative in which the Abrahamic God created the universe. In another type of story, the universe is created from the union of male and female deities, as in the Maori story of Rangi and Papa. In other stories, the universe is created by crafting it from pre-existing materials, such as the corpse of a dead god—as from Tiamat in the Babylonian epic "Enuma Elish" or from the giant Ymir in Norse mythology—or from chaotic materials, as in Izanagi and Izanami in Japanese mythology. In other stories, the universe emanates from fundamental principles, such as Brahman and Prakrti, the creation myth of the Serers, or the yin and yang of the Tao. | https://en.wikipedia.org/wiki?curid=31880 |
13,502 | The pre-Socratic Greek philosophers and Indian philosophers developed some of the earliest philosophical concepts of the universe. The earliest Greek philosophers noted that appearances can be deceiving, and sought to understand the underlying reality behind the appearances. In particular, they noted the ability of matter to change forms (e.g., ice to water to steam) and several philosophers proposed that all the physical materials in the world are different forms of a single primordial material, or "arche". The first to do so was Thales, who proposed this material to be water. Thales' student, Anaximander, proposed that everything came from the limitless "apeiron". Anaximenes proposed the primordial material to be air on account of its perceived attractive and repulsive qualities that cause the "arche" to condense or dissociate into different forms. Anaxagoras proposed the principle of "Nous" (Mind), while Heraclitus proposed fire (and spoke of "logos"). Empedocles proposed the elements to be earth, water, air and fire. His four-element model became very popular. Like Pythagoras, Plato believed that all things were composed of number, with Empedocles' elements taking the form of the Platonic solids. Democritus, and later philosophers—most notably Leucippus—proposed that the universe is composed of indivisible atoms moving through a void (vacuum), although Aristotle did not believe that to be feasible because air, like water, offers resistance to motion. Air will immediately rush in to fill a void, and moreover, without resistance, it would do so indefinitely fast. | https://en.wikipedia.org/wiki?curid=31880 |
13,503 | Although Heraclitus argued for eternal change, his contemporary Parmenides made the radical suggestion that all change is an illusion, that the true underlying reality is eternally unchanging and of a single nature. Parmenides denoted this reality as (The One). Parmenides' idea seemed implausible to many Greeks, but his student Zeno of Elea challenged them with several famous paradoxes. Aristotle responded to these paradoxes by developing the notion of a potential countable infinity, as well as the infinitely divisible continuum. Unlike the eternal and unchanging cycles of time, he believed that the world is bounded by the celestial spheres and that cumulative stellar magnitude is only finitely multiplicative. | https://en.wikipedia.org/wiki?curid=31880 |
13,504 | The Indian philosopher Kanada, founder of the Vaisheshika school, developed a notion of atomism and proposed that light and heat were varieties of the same substance. In the 5th century AD, the Buddhist atomist philosopher Dignāga proposed atoms to be point-sized, durationless, and made of energy. They denied the existence of substantial matter and proposed that movement consisted of momentary flashes of a stream of energy. | https://en.wikipedia.org/wiki?curid=31880 |
13,505 | The notion of temporal finitism was inspired by the doctrine of creation shared by the three Abrahamic religions: Judaism, Christianity and Islam. The Christian philosopher, John Philoponus, presented the philosophical arguments against the ancient Greek notion of an infinite past and future. Philoponus' arguments against an infinite past were used by the early Muslim philosopher, Al-Kindi (Alkindus); the Jewish philosopher, Saadia Gaon (Saadia ben Joseph); and the Muslim theologian, Al-Ghazali (Algazel). | https://en.wikipedia.org/wiki?curid=31880 |
13,506 | Astronomical models of the universe were proposed soon after astronomy began with the Babylonian astronomers, who viewed the universe as a flat disk floating in the ocean, and this forms the premise for early Greek maps like those of Anaximander and Hecataeus of Miletus. | https://en.wikipedia.org/wiki?curid=31880 |
13,507 | Later Greek philosophers, observing the motions of the heavenly bodies, were concerned with developing models of the universe-based more profoundly on empirical evidence. The first coherent model was proposed by Eudoxus of Cnidos. According to Aristotle's physical interpretation of the model, celestial spheres eternally rotate with uniform motion around a stationary Earth. Normal matter is entirely contained within the terrestrial sphere. | https://en.wikipedia.org/wiki?curid=31880 |
13,508 | "De Mundo" (composed before 250 BC or between 350 and 200 BC), stated, "Five elements, situated in spheres in five regions, the less being in each case surrounded by the greater—namely, earth surrounded by water, water by air, air by fire, and fire by ether—make up the whole universe". | https://en.wikipedia.org/wiki?curid=31880 |
13,509 | This model was also refined by Callippus and after concentric spheres were abandoned, it was brought into nearly perfect agreement with astronomical observations by Ptolemy. The success of such a model is largely due to the mathematical fact that any function (such as the position of a planet) can be decomposed into a set of circular functions (the Fourier modes). Other Greek scientists, such as the Pythagorean philosopher Philolaus, postulated (according to Stobaeus account) that at the center of the universe was a "central fire" around which the Earth, Sun, Moon and planets revolved in uniform circular motion. | https://en.wikipedia.org/wiki?curid=31880 |
13,510 | The Greek astronomer Aristarchus of Samos was the first known individual to propose a heliocentric model of the universe. Though the original text has been lost, a reference in Archimedes' book "The Sand Reckoner" describes Aristarchus's heliocentric model. Archimedes wrote: | https://en.wikipedia.org/wiki?curid=31880 |
13,511 | You, King Gelon, are aware the universe is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the universe just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface | https://en.wikipedia.org/wiki?curid=31880 |
13,512 | Aristarchus thus believed the stars to be very far away, and saw this as the reason why stellar parallax had not been observed, that is, the stars had not been observed to move relative each other as the Earth moved around the Sun. The stars are in fact much farther away than the distance that was generally assumed in ancient times, which is why stellar parallax is only detectable with precision instruments. The geocentric model, consistent with planetary parallax, was assumed to be an explanation for the unobservability of the parallel phenomenon, stellar parallax. The rejection of the heliocentric view was apparently quite strong, as the following passage from Plutarch suggests ("On the Apparent Face in the Orb of the Moon"): | https://en.wikipedia.org/wiki?curid=31880 |
13,513 | Cleanthes [a contemporary of Aristarchus and head of the Stoics] thought it was the duty of the Greeks to indict Aristarchus of Samos on the charge of impiety for putting in motion the Hearth of the Universe [i.e. the Earth], ... supposing the heaven to remain at rest and the Earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis | https://en.wikipedia.org/wiki?curid=31880 |
13,514 | The only other astronomer from antiquity known by name who supported Aristarchus's heliocentric model was Seleucus of Seleucia, a Hellenistic astronomer who lived a century after Aristarchus. According to Plutarch, Seleucus was the first to prove the heliocentric system through reasoning, but it is not known what arguments he used. Seleucus' arguments for a heliocentric cosmology were probably related to the phenomenon of tides. According to Strabo (1.1.9), Seleucus was the first to state that the tides are due to the attraction of the Moon, and that the height of the tides depends on the Moon's position relative to the Sun. Alternatively, he may have proved heliocentricity by determining the constants of a geometric model for it, and by developing methods to compute planetary positions using this model, like what Nicolaus Copernicus later did in the 16th century. During the Middle Ages, heliocentric models were also proposed by the Indian astronomer Aryabhata, and by the Persian astronomers Albumasar and Al-Sijzi. | https://en.wikipedia.org/wiki?curid=31880 |
13,515 | The Aristotelian model was accepted in the Western world for roughly two millennia, until Copernicus revived Aristarchus's perspective that the astronomical data could be explained more plausibly if the Earth rotated on its axis and if the Sun were placed at the center of the universe. | https://en.wikipedia.org/wiki?curid=31880 |
13,516 | As noted by Copernicus himself, the notion that the Earth rotates is very old, dating at least to Philolaus (c. 450 BC), Heraclides Ponticus (c. 350 BC) and Ecphantus the Pythagorean. Roughly a century before Copernicus, the Christian scholar Nicholas of Cusa also proposed that the Earth rotates on its axis in his book, "On Learned Ignorance" (1440). Al-Sijzi also proposed that the Earth rotates on its axis. Empirical evidence for the Earth's rotation on its axis, using the phenomenon of comets, was given by Tusi (1201–1274) and Ali Qushji (1403–1474). | https://en.wikipedia.org/wiki?curid=31880 |
13,517 | This cosmology was accepted by Isaac Newton, Christiaan Huygens and later scientists. Edmund Halley (1720) and Jean-Philippe de Chéseaux (1744) noted independently that the assumption of an infinite space filled uniformly with stars would lead to the prediction that the nighttime sky would be as bright as the Sun itself; this became known as Olbers' paradox in the 19th century. Newton believed that an infinite space uniformly filled with matter would cause infinite forces and instabilities causing the matter to be crushed inwards under its own gravity. This instability was clarified in 1902 by the Jeans instability criterion. One solution to these paradoxes is the Charlier Universe, in which the matter is arranged hierarchically (systems of orbiting bodies that are themselves orbiting in a larger system, "ad infinitum") in a fractal way such that the universe has a negligibly small overall density; such a cosmological model had also been proposed earlier in 1761 by Johann Heinrich Lambert. A significant astronomical advance of the 18th century was the realization by Thomas Wright, Immanuel Kant and others of nebulae. | https://en.wikipedia.org/wiki?curid=31880 |
13,518 | In 1919, when the Hooker Telescope was completed, the prevailing view still was that the universe consisted entirely of the Milky Way Galaxy. Using the Hooker Telescope, Edwin Hubble identified Cepheid variables in several spiral nebulae and in 1922–1923 proved conclusively that Andromeda Nebula and Triangulum among others, were entire galaxies outside our own, thus proving that universe consists of a multitude of galaxies. | https://en.wikipedia.org/wiki?curid=31880 |
13,519 | The modern era of physical cosmology began in 1917, when Albert Einstein first applied his general theory of relativity to model the structure and dynamics of the universe. | https://en.wikipedia.org/wiki?curid=31880 |
13,520 | In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson (; ). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume. | https://en.wikipedia.org/wiki?curid=23009144 |
13,521 | For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. | https://en.wikipedia.org/wiki?curid=23009144 |
13,522 | Another example is the number of decay events that occur from a radioactive source during a defined observation period. | https://en.wikipedia.org/wiki?curid=23009144 |
13,523 | The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work "Recherches sur la probabilité des jugements en matière criminelle et en matière civile" (1837). The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in "De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus" . This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre. | https://en.wikipedia.org/wiki?curid=23009144 |
13,524 | In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space. | https://en.wikipedia.org/wiki?curid=23009144 |
13,525 | A further practical application of this distribution was made by Ladislaus Bortkiewicz in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks; this experiment introduced the Poisson distribution to the field of reliability engineering. | https://en.wikipedia.org/wiki?curid=23009144 |
13,526 | A discrete random variable is said to have a Poisson distribution, with parameter formula_21 if it has a probability mass function given by: | https://en.wikipedia.org/wiki?curid=23009144 |
13,527 | The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution. | https://en.wikipedia.org/wiki?curid=23009144 |
13,528 | The equation can be adapted if, instead of the average number of events formula_26 we are given the average rate formula_27 at which events occur. Then formula_28 and: | https://en.wikipedia.org/wiki?curid=23009144 |
13,529 | If these conditions are true, then is a Poisson random variable, and the distribution of is a Poisson distribution. | https://en.wikipedia.org/wiki?curid=23009144 |
13,530 | The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals divided by the number of trials, as the number of trials approaches infinity (see Related distributions). | https://en.wikipedia.org/wiki?curid=23009144 |
13,531 | On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100 year interval, assuming the Poisson model is appropriate. | https://en.wikipedia.org/wiki?curid=23009144 |
13,532 | María Dolores Ugarte and colleagues report that the average number of goals in a World Cup soccer match is approximately 2.5 and the Poisson model is appropriate. | https://en.wikipedia.org/wiki?curid=23009144 |
13,533 | Suppose that astronomers estimate that large meteorites (above a certain size) hit the earth on average once every 100 years ( event per 100 years), and that the number of meteorite hits follows a Poisson distribution. What is the probability of meteorite hits in the next 100 years? | https://en.wikipedia.org/wiki?curid=23009144 |
13,534 | Under these assumptions, the probability that no large meteorites hit the earth in the next 100 years is roughly 0.37. The remaining is the probability of 1, 2, 3, or more large meteorite hits in the next 100 years. | https://en.wikipedia.org/wiki?curid=23009144 |
13,535 | In an example above, an overflow flood occurred once every 100 years The probability of no overflow floods in 100 years was roughly 0.37, by the same calculation. | https://en.wikipedia.org/wiki?curid=23009144 |
13,536 | In general, if an event occurs on average once per interval ( = 1), and the events follow a Poisson distribution, then In addition, as shown in the table for overflow floods. | https://en.wikipedia.org/wiki?curid=23009144 |
13,537 | The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups). The non-constant arrival rate may be modeled as a mixed Poisson distribution, and the arrival of groups rather than individual students as a compound Poisson process. | https://en.wikipedia.org/wiki?curid=23009144 |
13,538 | The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution, if one large earthquake increases the probability of aftershocks of similar magnitude. | https://en.wikipedia.org/wiki?curid=23009144 |
13,539 | Examples in which at least one event is guaranteed are not Poisson distributed; but may be modeled using a zero-truncated Poisson distribution. | https://en.wikipedia.org/wiki?curid=23009144 |
13,540 | Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a zero-inflated model. | https://en.wikipedia.org/wiki?curid=23009144 |
13,541 | The higher non-centered moments, of the Poisson distribution, are Touchard polynomials in : formula_44 where the {braces} denote Stirling numbers of the second kind. The coefficients of the polynomials have a combinatorial meaning. In fact, when the expected value of the Poisson distribution is 1, then Dobinski's formula says that the ‑th moment equals the number of partitions of a set of size . | https://en.wikipedia.org/wiki?curid=23009144 |
13,542 | If formula_46 for formula_47 are independent, then formula_48 A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables. | https://en.wikipedia.org/wiki?curid=23009144 |
13,543 | The lower bound can be proved by noting that formula_69 is the probability that formula_70 where formula_71 which is bounded below by formula_72 where formula_73 is relative entropy (See the entry on bounds on tails of binomial distributions for details). Further noting that formula_74 and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath "et al.". | https://en.wikipedia.org/wiki?curid=23009144 |
13,544 | The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore, it can be used as an approximation of the binomial distribution if is sufficiently large and "p" is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if is at least 20 and "p" is smaller than or equal to 0.05, and an excellent approximation if ≥ 100 and ≤ 10. formula_75 | https://en.wikipedia.org/wiki?curid=23009144 |
13,545 | The factor of formula_114 can be replaced by 2 if formula_115 is further assumed to be monotonically increasing or decreasing. | https://en.wikipedia.org/wiki?curid=23009144 |
13,546 | This distribution has been extended to the bivariate case. The generating function for this distribution is | https://en.wikipedia.org/wiki?curid=23009144 |
13,547 | The marginal distributions are Poisson("θ") and Poisson("θ") and the correlation coefficient is limited to the range | https://en.wikipedia.org/wiki?curid=23009144 |
13,548 | A simple way to generate a bivariate Poisson distribution formula_119 is to take three independent Poisson distributions formula_120 with means formula_121 and then set formula_122 The probability function of the bivariate Poisson distribution is | https://en.wikipedia.org/wiki?curid=23009144 |
13,549 | The free Poisson distribution with jump size formula_124 and rate formula_8 arises in free probability theory as the limit of repeated free convolution | https://en.wikipedia.org/wiki?curid=23009144 |
13,550 | In other words, let formula_127 be random variables so that formula_127 has value formula_124 with probability formula_130 and value 0 with the remaining probability. Assume also that the family formula_131 are freely independent. Then the limit as formula_132 of the law of formula_133 is given by the Free Poisson law with parameters formula_134 | https://en.wikipedia.org/wiki?curid=23009144 |
13,551 | This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process. | https://en.wikipedia.org/wiki?curid=23009144 |
13,552 | This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to formula_138 | https://en.wikipedia.org/wiki?curid=23009144 |
13,553 | We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book "Lectures on the Combinatorics of Free Probability" by A. Nica and R. Speicher | https://en.wikipedia.org/wiki?curid=23009144 |
13,554 | Poisson's probability mass function formula_143 can be expressed in a form similar to the product distribution of a Weibull distribution and a variant form of the stable count distribution. | https://en.wikipedia.org/wiki?curid=23009144 |
13,555 | The variable formula_144 can be regarded as inverse of Lévy's stability parameter in the stable count distribution: | https://en.wikipedia.org/wiki?curid=23009144 |
13,556 | where formula_146 is a standard stable count distribution of shape formula_147 and formula_148 is a standard Weibull distribution of shape formula_149 | https://en.wikipedia.org/wiki?curid=23009144 |
13,557 | Given a sample of measured values formula_150 for we wish to estimate the value of the parameter of the Poisson population from which the sample was drawn. The maximum likelihood estimate is | https://en.wikipedia.org/wiki?curid=23009144 |
13,558 | Since each observation has expectation so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of . It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB). Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for . | https://en.wikipedia.org/wiki?curid=23009144 |
13,559 | To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample formula_152 (called formula_153) and one that depends on the parameter formula_8 and the sample formula_152 only through the function formula_156 Then formula_157 is a sufficient statistic for formula_158 | https://en.wikipedia.org/wiki?curid=23009144 |
13,560 | The first term, formula_160 depends only on formula_161 The second term, formula_162 depends on the sample only through formula_163 Thus, formula_157 is sufficient. | https://en.wikipedia.org/wiki?curid=23009144 |
13,561 | To find the parameter that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function: | https://en.wikipedia.org/wiki?curid=23009144 |
13,562 | So is the average of the values. Obtaining the sign of the second derivative of "L" at the stationary point will determine what kind of extreme value is. | https://en.wikipedia.org/wiki?curid=23009144 |
13,563 | which is the negative of times the reciprocal of the average of the k. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function. | https://en.wikipedia.org/wiki?curid=23009144 |
13,564 | For completeness, a family of distributions is said to be complete if and only if formula_171 implies that formula_172 for all formula_158 If the individual formula_174 are iid formula_175 then formula_176 Knowing the distribution we want to investigate, it is easy to see that the statistic is complete. | https://en.wikipedia.org/wiki?curid=23009144 |
13,565 | For this equality to hold, formula_178 must be 0. This follows from the fact that none of the other terms will be 0 for all formula_179 in the sum and for all possible values of formula_158 Hence, formula_181 for all formula_8 implies that formula_183 and the statistic has been shown to be complete. | https://en.wikipedia.org/wiki?curid=23009144 |
13,566 | The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation from a Poisson distribution with mean "μ", a confidence interval for "μ" with confidence level is | https://en.wikipedia.org/wiki?curid=23009144 |
13,567 | where formula_186 is the quantile function (corresponding to a lower tail area "p") of the chi-squared distribution with degrees of freedom and formula_187 is the quantile function of a gamma distribution with shape parameter n and scale parameter 1. This interval is 'exact' in the sense that its coverage probability is never less than the nominal . | https://en.wikipedia.org/wiki?curid=23009144 |
13,568 | When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation): | https://en.wikipedia.org/wiki?curid=23009144 |
13,569 | For application of these formulae in the same context as above (given a sample of measured values each drawn from a Poisson distribution with mean ), one would set | https://en.wikipedia.org/wiki?curid=23009144 |
13,570 | In Bayesian inference, the conjugate prior for the rate parameter of the Poisson distribution is the gamma distribution. Let | https://en.wikipedia.org/wiki?curid=23009144 |
13,571 | denote that is distributed according to the gamma density "g" parameterized in terms of a shape parameter "α" and an inverse scale parameter "β": | https://en.wikipedia.org/wiki?curid=23009144 |
13,572 | Then, given the same sample of measured values as before, and a prior of Gamma("α", "β"), the posterior distribution is | https://en.wikipedia.org/wiki?curid=23009144 |
13,573 | It can be shown that gamma distribution is the only prior that induces linearity of the conditional mean. Moreover, a converse result exists which states that if the conditional mean is close to a linear function in the formula_195 distance than the prior distribution of must be close to gamma distribution in Levy distance. | https://en.wikipedia.org/wiki?curid=23009144 |
13,574 | The posterior mean E[] approaches the maximum likelihood estimate formula_196 in the limit as formula_197 which follows immediately from the general expression of the mean of the gamma distribution. | https://en.wikipedia.org/wiki?curid=23009144 |
13,575 | The posterior predictive distribution for a single additional observation is a negative binomial distribution, sometimes called a gamma–Poisson distribution. | https://en.wikipedia.org/wiki?curid=23009144 |
13,576 | Suppose formula_198 is a set of independent random variables from a set of formula_199 Poisson distributions, each with a parameter formula_200 formula_201 and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss formula_202 when formula_203 then, similar as in Stein's example for the Normal means, the MLE estimator formula_204 is inadmissible. | https://en.wikipedia.org/wiki?curid=23009144 |
13,577 | The Poisson distribution arises in connection with Poisson processes. It applies to various phenomena of discrete properties (that is, those that may happen 0, 1, 2, 3, … times during a given period of time or in a given area) whenever the probability of the phenomenon happening is constant in time or space. Examples of events that may be modelled as a Poisson distribution include: | https://en.wikipedia.org/wiki?curid=23009144 |
13,578 | Gallagher showed in 1976 that the counts of prime numbers in short intervals obey a Poisson distribution provided a certain version of the unproved prime r-tuple conjecture of Hardy-Littlewood is true. | https://en.wikipedia.org/wiki?curid=23009144 |
13,579 | The rate of an event is related to the probability of an event occurring in some small subinterval (of time, space or otherwise). In the case of the Poisson distribution, one assumes that there exists a small enough subinterval for which the probability of an event occurring twice is "negligible". With this assumption one can derive the Poisson distribution from the Binomial one, given only the information of expected number of total events in the whole interval. | https://en.wikipedia.org/wiki?curid=23009144 |
13,580 | Let the total number of events in the whole interval be denoted by formula_158 Divide the whole interval into formula_209 subintervals formula_210 of equal size, such that formula_211 (since we are interested in only very small portions of the interval this assumption is meaningful). This means that the expected number of events in each of the subintervals is equal to formula_212 | https://en.wikipedia.org/wiki?curid=23009144 |
13,581 | Now we assume that the occurrence of an event in the whole interval can be seen as a sequence of Bernoulli trials, where the formula_213-th Bernoulli trial corresponds to looking whether an event happens at the subinterval formula_214 with probability formula_212 The expected number of total events in formula_209 such trials would be formula_26 the expected number of total events in the whole interval. Hence for each subdivision of the interval we have approximated the occurrence of the event as a Bernoulli process of the form formula_218 As we have noted before we want to consider only very small subintervals. Therefore, we take the limit as formula_209 goes to infinity. | https://en.wikipedia.org/wiki?curid=23009144 |
13,582 | In this case the binomial distribution converges to what is known as the Poisson distribution by the Poisson limit theorem. | https://en.wikipedia.org/wiki?curid=23009144 |
13,583 | In several of the above examples — such as, the number of mutations in a given sequence of DNA—the events being counted are actually the outcomes of discrete trials, and would more precisely be modelled using the binomial distribution, that is | https://en.wikipedia.org/wiki?curid=23009144 |
13,584 | In such cases is very large and is very small (and so the expectation is of intermediate magnitude). Then the distribution may be approximated by the less cumbersome Poisson distribution formula_221 | https://en.wikipedia.org/wiki?curid=23009144 |
13,585 | This approximation is sometimes known as the "law of rare events", since each of the individual Bernoulli events rarely occurs. | https://en.wikipedia.org/wiki?curid=23009144 |
13,586 | The name "law of rare events" may be misleading because the total count of success events in a Poisson process need not be rare if the parameter is not small. For example, the number of telephone calls to a busy switchboard in one hour follows a Poisson distribution with the events appearing frequent to the operator, but they are rare from the point of view of the average member of the population who is very unlikely to make a call to that switchboard in that hour. | https://en.wikipedia.org/wiki?curid=23009144 |
13,587 | The variance of the binomial distribution is 1 − "p" times that of the Poisson distribution, so almost equal when "p" is very small. | https://en.wikipedia.org/wiki?curid=23009144 |
13,588 | The word "law" is sometimes used as a synonym of probability distribution, and "convergence in law" means "convergence in distribution". 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. "The Law of Small Numbers" is a book by Ladislaus Bortkiewicz about the Poisson distribution, published in 1898. | https://en.wikipedia.org/wiki?curid=23009144 |
13,589 | The Poisson distribution arises as the number of points of a Poisson point process located in some finite region. More specifically, if "D" is some region space, for example Euclidean space R, for which |"D"|, the area, volume or, more generally, the Lebesgue measure of the region is finite, and if denotes the number of points in "D", then | https://en.wikipedia.org/wiki?curid=23009144 |
13,590 | Poisson regression and negative binomial regression are useful for analyses where the dependent (response) variable is the count of the number of events or occurrences in an interval. | https://en.wikipedia.org/wiki?curid=23009144 |
13,591 | In a Poisson process, the number of observed occurrences fluctuates about its mean with a standard deviation formula_223 These fluctuations are denoted as "Poisson noise" or (particularly in electronics) as "shot noise". | https://en.wikipedia.org/wiki?curid=23009144 |
13,592 | The correlation of the mean and standard deviation in counting independent discrete occurrences is useful scientifically. By monitoring how the fluctuations vary with the mean signal, one can estimate the contribution of a single occurrence, "even if that contribution is too small to be detected directly". For example, the charge "e" on an electron can be estimated by correlating the magnitude of an electric current with its shot noise. If "N" electrons pass a point in a given time "t" on the average, the mean current is formula_224; since the current fluctuations should be of the order formula_225 (i.e., the standard deviation of the Poisson process), the charge formula_226 can be estimated from the ratio formula_227 | https://en.wikipedia.org/wiki?curid=23009144 |
13,593 | An everyday example is the graininess that appears as photographs are enlarged; the graininess is due to Poisson fluctuations in the number of reduced silver grains, not to the individual grains themselves. By correlating the graininess with the degree of enlargement, one can estimate the contribution of an individual grain (which is otherwise too small to be seen unaided). Many other molecular applications of Poisson noise have been developed, e.g., estimating the number density of receptor molecules in a cell membrane. | https://en.wikipedia.org/wiki?curid=23009144 |
13,594 | The Poisson distribution poses two different tasks for dedicated software libraries: "evaluating" the distribution formula_229, and "drawing random numbers" according to that distribution. | https://en.wikipedia.org/wiki?curid=23009144 |
13,595 | Computing formula_229 for given formula_231 and formula_8 is a trivial task that can be accomplished by using the standard definition of formula_229 in terms of exponential, power, and factorial functions. However, the conventional definition of the Poisson distribution contains two terms that can easily overflow on computers: and . The fraction of to ! can also produce a rounding error that is very large compared to "e", and therefore give an erroneous result. For numerical stability the Poisson probability mass function should therefore be evaluated as | https://en.wikipedia.org/wiki?curid=23009144 |
13,596 | which is mathematically equivalent but numerically stable. The natural logarithm of the Gamma function can be obtained using the codice_1 function in the C standard library (C99 version) or R, the codice_2 function in MATLAB or SciPy, or the codice_3 function in Fortran 2008 and later. | https://en.wikipedia.org/wiki?curid=23009144 |
13,597 | The less trivial task is to draw integer random variate from the Poisson distribution with given formula_158 | https://en.wikipedia.org/wiki?curid=23009144 |
13,598 | A simple algorithm to generate random Poisson-distributed numbers (pseudo-random number sampling) has been given by Knuth: | https://en.wikipedia.org/wiki?curid=23009144 |
13,599 | The complexity is linear in the returned value , which is on average. There are many other algorithms to improve this. Some are given in Ahrens & Dieter, see below. | https://en.wikipedia.org/wiki?curid=23009144 |
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