Concept of Randomness in Statistics

Concept of second in StatisticsPart I instauration macrocosm on Freshman SeminarFreshman seminar 1205M offers great opportunities for students to work intimately with professors from the acquirement faculty on various areas of mathematics. The seminar was targeted to encourage us to open our minds to seminal ideas and develop curiosity of influential mathematical theories and various subgroups of contemporary mathematics. In addition to exposure to selected subtopics in contemporary mathematics, we had valuable opportunities to develop our manifestation and academic essay writing skills.1.2 Important roles of Analogy and IntuitionThe diachronic development of mathematics is signifi savet jointtly crookd by intuition acquired from real feeling experience and analogy quoted from various former(a)wise areas (Harrison Treagust, 1993). Analogy is an extraordinary method acting in developing brand-new concepts in the hi history of science. In this module, renowned topics in th e contemporary mathematics, including geometry, number surmise, set theory, sulphur and game theory have been discussed. Among all topics, our team worked on Analogy and Intuition of Randomness. In this seminar, various creative analogy ideas and intuition/counter-intuition thinking have been presented establish on specific cases in unexampled mathematics.1.3 Method on enquiry and PresentationOur team collected relevant source materials on the haphazardness, including books, journals, and websites on the Internet. As for presenting applications of haphazardness, in particular, we focused on the historical development of randomness theory, the simplified key concepts in randomness, the counter-intuitive stories happened, all overlapping with early(a) fields in nature, and some signifi cant and influential applications of randomness theory in our routine life. We omitted complicated theories, skilful formulas and rigorous proofs. Throughout the whole semester, our team has co nducted deuce cosy presentations on randomness. In order to illustrate randomness clear and intuitively, we adopted various methods problem solving, in-class quizzes, presentations and attractive stories. Subtopics included Biology, quantum physics, finance, audio engineering, statistics and so on.Part II Report on Randomness2.1 Randomness on Communication opening2.1.1 Introduction of Noise in Communication TheoryIn statistics, irrelevant or meaningless data is considered sound (random error). Whereas in communication theory, random disturbance in a house is called folie. In essence, go consists of a large number of disturbances with a statistically disarrange time distribution.It is assumed that hinderance signals have ability spectral slow-wittedness that is proportional to 1/f, where f stands for frequencies of hoo-ha. For example, the spectral density of albumin incumbrance is = 0, while sound disturbance has = 1. This special character is wide used for disti nguishing among tincts of haphazardness.2.1.2 Laws and Criterions Used to Distinguish Colors and Characteristics of NoiseThe color names for intervention are derived from an analogy between the spectrum of preventive and the resembling spectrum of lights with different visible modify. For instance, if we translate the sound wave of ovalbumin encumbrance into light waves, the resulting light pass on be viewed as white color. In electronics, physics, and many other areas, the color of a noise signal is normally understood as some characteristics of its actor spectrum. As different colors of noise have significantly different properties. Therefore, each pattern of noise requires a specific color to match with it.Start with the most well-known(a) one White noise, people name different noise afterwards(prenominal) colors. This is in analogy with white color light, which has a flat spectrum of power on its frequency range. Other colors, such(prenominal) as violet, blue, red, pink, are and then given to different noises with extremely similar spectrum characteristics.Although most of them have the like noise patterns with specific disciplines, there are also plenty of noise spectrums with imprecise and informal definitions, like black noise, green noise, brown noise and so on.These below parts were summarized from Wikipedia terms Noise (electronics)Sites http//en.wikipedia.org/wiki/Noise_%28electronics%292.1.3 versed Sources of NoiseThermal noise is generated from the random thermal drive of charges (usually electrons) deep down electrical conductors. The amplitude of the signal has a probability density depart similar to the Gaussian (Normal) distribution. The amplitude of thermal noise depends on the temperature of the circuit. pass noise results from unavoidable random fluctuations when the charges (such as electrons) jump over a gap inside the electric circuits. It sounds rather similar to the noise created by rain falling on a tin roof.Flick er noise has a frequency spectrum that falls down into the gameyer frequencies areas steadily.Burst noise consists of sudden step-like transitions between two or more than levels at random and un call offable times. It sounds like eating popcorn.2.1.4 Outer sources of NoiseAtmospheric noise is the natural disturbance caused by electricity discharges in thunderstorm and other natural disturbances occurring in nature, like disruptions of high-voltage wires.Industrial noises are produced by automobiles, aircrafts and so on. The disturbances are produced by the discharge processes in these operations as well, which is similar to the atmospheric noise.Extraterrestrial noises come from the universe. These noises include Solar Noise, which is a radiation from the sun collectible to its intense nuclear reactions and the consequent high temperature, and Cosmic Noise, which are able to transmit its radiation and cosmic rays to more or less anywhere.2.1.5 Classification of Different Color s of NoiseThis part was adapted and summarized from an online introductory article White, pink, blue and violet The colors of noise from the Wired powder magazine Science Column, Author Duncan Geere, Date Apr. 07, 2011White noiseWhite noise has a constant power distribution density on its spectrum. It is named after the white color light, which has a flatten frequency everywhere on the spectrum. The term is widely applied in many scientific and skillful areas, including physics, audio engineering, telecommunications, statistical forecasting and many other areas. Specifically, White noise is used as a generator for random numbers. In addition, abide forecasting websites also use white noise to generate random digit patterns and simulate real weather.Pink noiseThe power density of pink noise decreases proportionally to 1/f. In the ultimo, the term of electric discharge noise sometimes refers to pink noise, but it will be more appropriate if we strictly apply it only to electron ic circuits. Moreover, Pink noise is also used in analysis of meteorological data and railroad siding radiation power of some astronomical bodies.Brown noise tally to the precise definition, the term Brown noise refers to a noise whose power density decreases inversely proportional to f2.The density function can be generated from integrating white noise or via an algorithm of Brownian motion simulation. Brown noise is not named after the color brown spectrum, which is diaphanous from other noises. It can be used in climatology to constitute temper shifts. However, within the scientific community, scientists have been arguing about its value for such purposes for a long time. patrician noiseThe power density of Blue noise is proportional to frequency. Blue noise has an increasing frequency over a finite frequency range. Blue noise is similar to pink noise, but instead of a decreasing spectrum, we observe an increasing one. sometimes it is mixed up with reddish blue noise in infor mal discussion.Violet noiseViolet noise is also known as the empurpled noise. The power density of Violet noise is proportional to f2, which convey it increases in quadratic form. Violet noise is like another translation of Brownian noise. Moreover, as Violet noise is the result of differentiating the white noise signal density, so people also call it the Differentiated White noise.Grey noiseGrey noise is a special kind of white noise process with characteristic equal loudness curve. However, it has a higher power density at both ends of the frequency spectrum but very little power near the center. Apparently, this is different from the standard white noise which is equal loud across its power density. However, actually this phenomenon is due to the humans hearing illusion.2.2 Randomness on Finance2.2.1 Brief Introduction to Efficient trade HypothesisThis part was summarized based on an online informal introductory article The Efficient foodstuffs Hypothesis, Authors Jonathan C larks, Tomas Jandik, Gershon Mandelker, Website www.e-m-h.orgIn financial fields, the efficient- commercialise scheme asserts that stock market prices will evolve with respect to to a random walk. They have the same probability distribution and independent of each other. Random walk states that stocks take a random and unpredictable path. The probability of a stocks future price going up is equal to going down. Therefore, the past movement (or trend) of a specific stock price or the overall market performance cannot be used as the basis to predict future movements. In addition, it is impossible to outperform the entire market without fetching additional risk or putting extra efforts. However, EMH proves that a long-term buy-and-hold strategy is the most efficient, because long term prices will approximately smooth performance of the federation very well, whereas short term movements in prices can be only described as a random walk.2.2.2 historic Backgrounds of Efficient bollo cks upket HypothesisThis part was summarized based on an online nonprofit educational website www.e-m-h.org and a research paper History of the Efficient rapeket Hypothesis, Nov.2004, Author ruintin, Sewell, Publisher University College London.Historically, the randomness of stock market prices was firstly poserled by a French broker, Jules Regnault, in 1863. Shortly after, a French mathematician, Louis Bachelier, real the mathematics of Brownian motion in 1900. In 1923, the famous economist, Keynes clearly give tongue to that investors in financial markets would be rewarded not for knowing better than other participants in the market, but rather for risk taking.After the WWII, the efficient-market guesswork emerged as an outstanding theory in the mid-1960s. In the 1960s, Mandelbrot proposed a randomness model for stock pricing. Fama discussed about Mandelbrots hypothesis and concluded that the market data confirmed his model. In addition, he defined the so-called efficient m arket for the first time, in his paper Random Walks in Stock Market Prices. He explained how random walks in stock market significantly influence individual stock prices. Later, he introduced definitions for three forms of financial market energy weak, semi-strong and strong.The term was eventually popularized when Burton Malkiel, a Professor of Economics at Princeton University, create his classic and prominent book A Random Walk eat Wall Street.2.2.3 Three Major Types of Markets Weak, Semi-Strong and StrongThe three types of EMH were summarized based on an online technical blog The Efficient Markets Hypothesis, Author Jodi Beggs, Website About.comWeak throw of strengthWe cannot predict future prices through analyzing prices from the past. And we cannot earn bare(a)ive returns by using information based on historical data. In this level, technical analysis is always profitable, as share prices exhibit no dependencies on their past. This implies that future prices depend entir ely on performance of companies.Semi-Strong Form of EfficiencyInformation other than market data is released, such as repetitive news, companies management, financial accounting reports, companies latest products. Under such condition, share prices will reflect the new information very rapidly. Therefore, investors cannot gain any redundance returns by trading on the public information. Semi-strong-form efficiency market implies that incomplete technical analysis nor fundamental analysis can produce excess returns.Strong Form of EfficiencyUnder such condition, information typically held by corporate insiders is released. Therefore, share prices reflect not only previously public information, but all private information as well. Theoretically, no one can earn excess returns. However, even before major changes are exposed to the public, corporate insiders are able to trade their companys stocks from abnormal profits. Fortunately, such insider trading is banned by inspection auth orities, like the Securities and Exchange Commission.2.2.4 Arguments and Critics on Efficiency Market HypothesisHowever, critics charge up that the theorys applications in markets results in financial crisis. In response, proponents of the hypothesis state that the theory is only a simplification model of the knowledge domain, which pith that it may not always hold true under every conditions. Hence, the market is only practically efficient for merely investment purposes in the real world rather than other aims.2.2.5 Interesting Counter-intuitive Stories on MonkeysThe story was adapted from the Forbes Magazine, Personal Finance Column, Author Rick Ferri, Date Dec, 20, 2012In order to verify the Efficient Market Hypothesis and illustrate the theories explicitly to the public, a group of researchers conducted a monkey experiment. They randomly picked up xxx stocks from a one thousand stocks poll and then let a hundred monkeys throw darts at the stocks printing on newspaper. They u nplowed repeating this experiment for five decades, and tracked the results.In the end, to their surprise, monkeys performance instant the index by 1.7% per year, which indicates that, there is certain situation where traditional technical analysis cannot even beat randomly-selected portfolios. The results have shocked the whole world by how greatly randomness affects the market stock prices.2.3 Randomness in natural philosophy and Biology2.3.1 Application of Randomness in Modern PhysicsIn the early 19th century, physicists use the philosophy of randomness to content motions and behaviors of molecules, and they build models in thermodynamics to explain phenomenon in gas experiments.In the twentieth century, when the era comes for quantum mechanics, microscopic phenomena are considered as completely random. Randomness of things like radioactive decay, photons passing through polarizers, and other bizarre quantum effects cannot be explained and predicted with classical theories in the usual way (Scott, 2009). Therefore, physicists propose a new theory, which claims that in a microscopic world, some of the outcomes come forth casual and random. For example, when we describe a radioactive atom, we cannot predict when the atom will decay. What only left for us is the probability of decay during a specific given period. In order to solve this mystery, Einstein postulates the Hidden Variable theory, which states that nature contains irreducible randomness properties and variables work beyond our scope somehow, but they actually agree the outcomes appear in our world.2.3.2 Application of Randomness in BiologyThe modern evolutionary states that the diversity of life is due to natural selection. Randomness, an essential voice of biological diversity, is associated with the growth of biological organization during evolution (Longo Montevil, 2012). It plays important roles in determining genetic mutation, and the significance of randomness effects appear at differ ent sizes, from microorganisms to large mammals (Bonner, 2013). During this process, a number of random genetic mutations appear in the gene library under both inner and other influences. Although this process is purely random, it indeed systematically leads to a higher peril for survival and reproduction of those individuals who possess these mutations than those without them. This mechanism plays crucial roles in the survivals of animals.Surprisingly, randomness in biology has remarkable relations to quantum physics. Schrodinger proposes his notion of negative reciprocal ohm as a form of Gibbs free energy, which also behaves similarly to randomness properties in abstract quantum world (Schrodinger, 1944).Part III ReferencesBeggs, J. (2014). The Efficient Markets Hypothesis. About. Retrieved Mar 30, 2014 from http//economics.about.com/od/Financial-Markets-Category/a/The-Efficient-Markets-Hypothesis.htmBonner, J. (2013). Randomness in Evolution. Princeton University Press. Retriev ed Mar 30, 2014 from http//press.princeton.edu/titles/9958.htmlClarke, J. Jandik, T. (2012). The Efficient Markets Hypothesis. Retrieved Mar 30, 2014 from http//ww.e-m-h.org/ClJM.pdfFerri, R. (2012). Any Monkey Can Beat The Market. Forbes. Retrieved Mar 30, 2014 from http//www.forbes.com/sites/rickferri/2012/12/20/any-monkey-can-beat-the-market/Geere, D. (2011). White, pink, blue and violet The colors of noise. Wired. Retrieved Mar 30, 2014 from http//www.wired.co.uk/news/archive/2011-04/7/colours-of-noise/viewallHarrison, A. G., Treagust, D. F. (1994). Science analogies. The Science Teacher, 61, 40-43.Longo, G Montevil, M. (2012). Randomness Increases Order in biologicalEvolution. Retrieved Mar 30, 2014 from http//www.researchgate.net/profile/Giuseppe_Longo2/publication/221350338_Randomness_Increases_Order_in_Biological_Evolution/file/60b7d51544f17cb8d8.pdfSchrodinger, E. What Is Life? Cambridge U.P. (1944)Scoot, J. (2009). Do physicists in reality believe in true randomness?As k a Mathematician. Retrieved Mar 30, 2014 from http//www.askamathematician.com/2009/12/q-do-physicists-really-believe-in-true-randomness/Sewell, M. (2004). History of the efficient market hypothesis. Retrieved Mar 30, 2014 from http//www.cs.ucl.ac.uk/fileadmin/UCL-CS/images/Research_Student_Information/RN_11_04.pdf