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Random Permutation
A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation. A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards. Generating random permutations Entry-by-entry brute force method One method of generating a random permutation of a set of length ''n'' uniformly at random (i.e., each of the ''n''! permutations is equally likely to appear) is to generate a sequence by taking a random number between 1 and ''n'' sequentially, ensuring that there is no repetition, and interpreting this sequence (''x''1, ..., ''x''''n'') as the permutation : \begin 1 & 2 & 3 & \cdots & n \\ x_1 & x_2 & x_3 & \cdots & x_n \\ \end, shown here in two-line notation. This brute-force method will require occasional retries whenever the ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy. The field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. Weisstein continuously improved the notes and accepted corrections and comments from online readers. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Permutation
In cryptography, a pseudorandom permutation (PRP) is a function that cannot be distinguished from a random permutation (that is, a permutation selected at random with uniform probability, from the family of all permutations on the function's domain) with practical effort. Definition Let ''F'' be a mapping \left\^n \times \left\^s \rightarrow \left\^n. ''F'' is a PRP if and only if * For any K \in \left\^s, F_K is a bijection from \left\^n to \left\^n, where F_K(x)=F(x,K). * For any K \in \left\^s, there is an "efficient" algorithm to evaluate F_K(x) for any x \in \left\^n,. * For all probabilistic polynomial-time distinguishers D: \left, Pr\left(D^(1^n) = 1\right) - Pr\left(D^(1^n) = 1\right) \ < \varepsilon(s), where is chosen uniformly at random and is chosen uniformly at random from the set of permutations on ''n''-bit strings. A pseudorandom permutation family is a collection of pseudorandom permutations, where a specific p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shuffle
Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome. __TOC__ Techniques Overhand One of the easiest shuffles to accomplish after a little practice is the overhand shuffle. Johan Jonasson wrote, "The overhand shuffle... is the shuffling technique where you gradually transfer the deck from, say, your right hand to your left hand by sliding off small packets from the top of the deck with your thumb." In detail as normally performed, with the pack initially held in the left hand (say), most of the cards are grasped as a group from the bottom of the pack between the thumb and fingers of the right hand and lifted clear of the small group that remains in the left hand. Small packets are then released from the right hand a packet at a time so that they drop on the top of the pack accumulating in the left hand. The proces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golomb–Dickman Constant
In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is :\lambda = 0.62432 99885 43550 87099 29363 83100 83724\dots It is not known whether this constant is rational or irrational. Definitions Let ''a''''n'' be the average — taken over all permutations of a set of size ''n'' — of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is : \lambda = \lim_ \frac. In the language of probability theory, \lambda n is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size ''n''. In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer. More precisely, :\lambda = \lim_ \frac1n \sum_^n \frac, where P_1(k) is the largest prime factor of ''k'' . So if ''k'' is a ''d'' digit integer, then \lambda d is the asymptotic average number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faro Shuffle
The faro shuffle (American), weave shuffle (British), or dovetail shuffle is a method of shuffling playing cards, in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table interleaved. Diaconis, Graham, and Kantor also call this the technique, when used in magic. Mathematicians use the term "faro shuffle" to describe a precise rearrangement of a deck into two equal piles of 26 cards which are then interwoven perfectly. Description A right-handed practitioner holds the cards from above in the left hand and from below in the right hand. The deck is separated into two preferably equal parts by simply lifting up half the cards with the right thumb slightly and pushing the left hand's packet forward away from the right hand. The two packets are often crossed and tapped against each other to align them. They are then pushed together on the short sides and bent either up or down. The cards will then alt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Genetics
Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, speciation, and population stratification, population structure. Population genetics was a vital ingredient in the emergence of the Modern synthesis (20th century), modern evolutionary synthesis. Its primary founders were Sewall Wright, J. B. S. Haldane and Ronald Fisher, who also laid the foundations for the related discipline of quantitative genetics. Traditionally a highly mathematical discipline, modern population genetics encompasses theoretical, laboratory, and field work. Population genetic models are used both for statistical inference from DNA sequence data and for proof/disproof of concept. What sets population genetics apart from newer, more phenotypic approaches to modelling evolution, such as evolutionary game theory and evo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ewens's Sampling Formula
In population genetics, Ewens's sampling formula, describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample. Definition Ewens's sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of ''n'' gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are ''a''1 alleles represented once in the sample, and ''a''2 alleles represented twice, and so on, is :\operatorname(a_1,\dots,a_n; \theta)=\prod_^n, for some positive number ''θ'' representing the population mutation rate, whenever a_1, \ldots, a_n is a sequence of nonnegative integers such that :a_1+2a_2+3a_3+\cdots+na_n=\sum_^ i a_i = n.\, The phrase "under certain conditions" used above is made precise by the following assumptions: * The sample size ''n'' is small by comparison to the size of the whole population; and * Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Permutation Statistics
The statistics of random permutations, such as the cycle structure of a random permutation are of fundamental importance in the analysis of algorithms, especially of sorting algorithms, which operate on random permutations. Suppose, for example, that we are using quickselect (a cousin of quicksort) to select a random element of a random permutation. Quickselect will perform a partial sort on the array, as it partitions the array according to the pivot. Hence a permutation will be less disordered after quickselect has been performed. The amount of disorder that remains may be analysed with generating functions. These generating functions depend in a fundamental way on the generating functions of random permutation statistics. Hence it is of vital importance to compute these generating functions. The article on random permutations contains an introduction to random permutations. The fundamental relation Permutations are sets of labelled cycles. Using the labelled case of the Fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diehard Tests
The diehard tests are a battery of statistical tests for measuring the quality of a random number generator. They were developed by George Marsaglia over several years and first published in 1995 on a CD-ROM of random numbers. Test overview ; Birthday spacings : Choose random points on a large interval. The spacings between the points should be asymptotically exponentially distributed. The name is based on the birthday paradox. ; Overlapping permutations : Analyze sequences of five consecutive random numbers. The 120 possible orderings should occur with statistically equal probability. ; Ranks of matrices : Select some number of bits from some number of random numbers to form a matrix over , then determine the rank of the matrix. Count the ranks. ; Monkey tests : Treat sequences of some number of bits as "words". Count the overlapping words in a stream. The number of "words" that do not appear should follow a known distribution. The name is based on the infinite monkey theorem. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomness Test
A randomness test (or test for randomness), in data evaluation, is a test used to analyze the distribution of a set of data to see if it can be described as random (patternless). In stochastic modeling, as in some computer simulations, the hoped-for randomness of potential input data can be verified, by a formal test for randomness, to show that the data are valid for use in simulation runs. In some cases, data reveals an obvious non-random pattern, as with so-called "runs in the data" (such as expecting random 0–9 but finding "4 3 2 1 0 4 3 2 1..." and rarely going above 4). If a selected set of data fails the tests, then parameters can be changed or other randomized data can be used which does pass the tests for randomness. Background The issue of randomness is an important philosophical and theoretical question. Tests for randomness can be used to determine whether a data set has a recognisable pattern, which would indicate that the process that generated it is significa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
