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Infinite Monkey Theorem
The infinite monkey theorem states that a monkey hitting keys independently and at randomness, random on a typewriter keyboard for an infinity, infinite amount of time will almost surely type any given text, including the complete works of William Shakespeare. More precisely, under the assumption of independence and randomness of each keystroke, the monkey would almost surely type every possible finite text an infinite number of times. The theorem can be generalized to state that any infinite sequence of independent events whose probabilities are uniformly bounded below by a positive number will almost surely have infinitely many occurrences. In this context, "almost surely" is a mathematical term meaning the event happens with probability 1, and the "monkey" is not an actual monkey, but a metaphor for an Abstract and concrete, abstract device that produces an endless random sequence of letters and symbols. Variants of the theorem include multiple and even infinitely many indepen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Chimpanzee Seated At Typewriter
The chimpanzee (; ''Pan troglodytes''), also simply known as the chimp, is a species of great ape native to the forests and savannahs of tropical Africa. It has four confirmed subspecies and a fifth proposed one. When its close relative the bonobo was more commonly known as the pygmy chimpanzee, this species was often called the common chimpanzee or the robust chimpanzee. The chimpanzee and the bonobo are the only species in the genus ''Pan''. Evidence from fossils and DNA sequencing shows that ''Pan'' is a sister taxon to the human lineage and is thus humans' closest living relative. The chimpanzee is covered in coarse black hair but has a bare face, fingers, toes, palms of the hands, and soles of the feet. It is larger and more robust than the bonobo, weighing for males and for females and standing . The chimpanzee lives in groups that range in size from 15 to 150 members, although individuals travel and forage in much smaller groups during the day. The species lives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Jonathan Swift
Jonathan Swift (30 November 1667 – 19 October 1745) was an Anglo-Irish writer, essayist, satirist, and Anglican cleric. In 1713, he became the Dean (Christianity), dean of St Patrick's Cathedral, Dublin, and was given the sobriquet "Dean Swift". His trademark deadpan and ironic style of writing, particularly in works such as ''A Modest Proposal'' (1729), has led to such satire being subsequently termed as "Swiftian". He wrote the satirical book ''Gulliver's Travels'' (1726), which became his best-known publication and popularised the fictional island of Lilliput and Blefuscu, Lilliput. Following the remarkable success of his works, Swift came to be regarded by many as the greatest satirist of the Georgian era, and one of the foremost prose satirists in the history of English literature. Swift also authored works such as ''A Tale of a Tub'' (1704) and ''An Argument Against Abolishing Christianity'' (1712). He originally published all of his works under pseudonyms—including L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Borel–Cantelli Lemma
In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first decades of the 20th century. A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will have probability of either zero or one. Accordingly, it is the best-known of a class of similar theorems, known as zero-one laws. Other examples include Kolmogorov's zero–one law and the Hewitt–Savage zero–one law. Statement of lemma for probability spaces Let ''E''1, ''E''2, ... be a sequence of events in some probability space. The Borel–Cantelli lemma states: Here, "lim sup" denotes limit supremum of the sequence of events. That is, lim sup ''E''''n'' is the outcome that infinitely many of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Substring
In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequence of "''It was the best of times''", but not a substring. Prefixes and suffixes are special cases of substrings. A prefix of a string S is a substring of S that occurs at the beginning of S; likewise, a suffix of a string S is a substring that occurs at the end of S. The substrings of the string "''apple''" would be: "''a''", "''ap''", "''app''", "''appl''", "''apple''", "''p''", "''pp''", "''ppl''", "''pple''", "''pl''", "''ple''", "''l''", "''le''" "''e''", "" (note the empty string at the end). Substring A string u is a substring (or factor) of a string t if there exists two strings p and s such that t = pus. In particular, the empty string is a substring of every string. Example: The string u=ana is equal to substrings (and subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Uniform Distribution (discrete)
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein each of some finite whole number ''n'' of outcome values are equally likely to be observed. Thus every one of the ''n'' outcome values has equal probability 1/''n''. Intuitively, a discrete uniform distribution is "a known, finite number of outcomes all equally likely to happen." A simple example of the discrete uniform distribution comes from throwing a fair six-sided die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of each given value is 1/6. If two dice were thrown and their values added, the possible sums would not have equal probability and so the distribution of sums of two dice rolls is not uniform. Although it is common to consider discrete uniform distributions over a contiguous range of integers, such as in this six-sided die example, one can define discrete uniform distributions over any finite set. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Alphabet
An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from another in a given language. Not all writing systems represent language in this way: a syllabary assigns symbols to spoken syllables, while logographies assign symbols to words, morphemes, or other semantic units. The first letters were invented in Ancient Egypt to serve as an aid in writing Egyptian hieroglyphs; these are referred to as Egyptian uniliteral signs by lexicographers. This system was used until the 5th century AD, and fundamentally differed by adding pronunciation hints to existing hieroglyphs that had previously carried no pronunciation information. Later on, these phonemic symbols also became used to transcribe foreign words. The first fully phonemic script was the Proto-Sinaitic script, also descending from Egyptian hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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String (computer Science)
In computer programming, a string is traditionally a sequence of character (computing), characters, either as a literal (computer programming), literal constant or as some kind of Variable (computer science), variable. The latter may allow its elements to be Immutable object, mutated and the length changed, or it may be fixed (after creation). A string is often implemented as an array data structure of bytes (or word (computer architecture), words) that stores a sequence of elements, typically characters, using some character encoding. More general, ''string'' may also denote a sequence (or List (abstract data type), list) of data other than just characters. Depending on the programming language and precise data type used, a variable (programming), variable declared to be a string may either cause storage in memory to be statically allocated for a predetermined maximum length or employ dynamic allocation to allow it to hold a variable number of elements. When a string appears lit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Limit Of A Function
Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function. Formal definitions, first devised in the early 19th century, are given below. Informally, a function assigns an output to every input . We say that the function has a limit at an input , if gets closer and closer to as moves closer and closer to . More specifically, the output value can be made ''arbitrarily'' close to if the input to is taken ''sufficiently'' close to . On the other hand, if some inputs very close to are taken to outputs that stay a fixed distance apart, then we say the limit ''does not exist''. The notion of a limit has many applications in modern calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Statistical Assumption
Statistics, like all mathematical disciplines, does not infer valid conclusions from nothing. Inferring interesting conclusions about real statistical populations almost always requires some background assumptions. Those assumptions must be made carefully, because incorrect assumptions can generate wildly inaccurate conclusions. Here are some examples of statistical assumptions: *Independence of observations from each other (this assumption is an especially common error). *Independence of observational error from potential confounding effects. *Exact or approximate normality of observations (or errors). *Linearity of graded responses to quantitative stimuli, e.g., in linear regression. Classes of assumptions There are two approaches to statistical inference: ''model-based inference'' and ''design-based inference''. Both approaches rely on some statistical model to represent the data-generating process. In the model-based approach, the model is taken to be initially unknown, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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San Francisco
San Francisco, officially the City and County of San Francisco, is a commercial, Financial District, San Francisco, financial, and Culture of San Francisco, cultural center of Northern California. With a population of 827,526 residents as of 2024, San Francisco is the List of California cities by population, fourth-most populous city in the U.S. state of California and the List of United States cities by population, 17th-most populous in the United States. San Francisco has a land area of at the upper end of the San Francisco Peninsula and is the County statistics of the United States, fifth-most densely populated U.S. county. Among U.S. cities proper with over 250,000 residents, San Francisco is ranked first by per capita income and sixth by aggregate income as of 2023. San Francisco anchors the Metropolitan statistical area#United States, 13th-most populous metropolitan statistical area in the U.S., with almost 4.6 million residents in 2023. The larger San Francisco Bay Area ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Earthquake
An earthquakealso called a quake, tremor, or tembloris the shaking of the Earth's surface resulting from a sudden release of energy in the lithosphere that creates seismic waves. Earthquakes can range in intensity, from those so weak they cannot be felt, to those violent enough to propel objects and people into the air, damage critical infrastructure, and wreak destruction across entire cities. The seismic activity of an area is the frequency, type, and size of earthquakes experienced over a particular time. The seismicity at a particular location in the Earth is the average rate of seismic energy release per unit volume. In its most general sense, the word ''earthquake'' is used to describe any seismic event that generates seismic waves. Earthquakes can occur naturally or be induced by human activities, such as mining, fracking, and nuclear weapons testing. The initial point of rupture is called the hypocenter or focus, while the ground level directly above it is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Moscow
Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents within the city limits, over 19.1 million residents in the urban area, and over 21.5 million residents in Moscow metropolitan area, its metropolitan area. The city covers an area of , while the urban area covers , and the metropolitan area covers over . Moscow is among the world's List of largest cities, largest cities, being the List of European cities by population within city limits, most populous city entirely in Europe, the largest List of urban areas in Europe, urban and List of metropolitan areas in Europe, metropolitan area in Europe, and the largest city by land area on the European continent. First documented in 1147, Moscow became the capital of the Grand Principality of Moscow, which led the unification of the Russian lan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |