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The Galton board, also known as a quincunx or bean machine, is a device for statistical experiments named after English scientist Sir Francis Galton. It consists of an upright board with evenly spaced nails (or pegs) driven into its upper half, where the nails are arranged in staggered order, and a lower half divided into a number of evenly-spaced rectangular slots. The front of the device is covered with a glass cover to allow viewing of both nails and slots. In the middle of the upper edge, there is a funnel into which balls can be poured, where the diameter of the balls must be much smaller than the distance between the nails. The funnel is located precisely above the central nail of the second row so that each ball, if perfectly centered, would fall vertically and directly onto the uppermost point of this nail's surface (Kozlov and Mitrofanova 2002). The figure above shows a variant of the board in which only the nails that can potentially be hit by a ball..

Let be a real symmetric matrix of large order having random elements that for are independently distributed with equal densities, equal second moments , and th moments bounded by constants independent of , , and . Further, let be the number of eigenvalues of that lie in the interval for real . Then(Wigner 1955, 1958). This law was first observed by Wigner (1955) for certain special classes of random matrices arising in quantum mechanical investigations.The distribution of eigenvalues of a symmetric random matrix with entries chosen from a standard normal distribution is illustrated above for a random matrix.Note that a large real symmetric matrix with random entries taken from a uniform distribution also obeys the semicircle law with the exception that it also possesses exactly one large eigenvalue...

Let be (possibly complex) eigenvalues of a set of random real matrices with entries independent and taken from a standard normal distribution. Then as , is uniformly distributed on the unit disk in the complex plane. For small , the distribution shows a concentration along the real line accompanied by a slight paucity above and below (with interesting embedded structure). However, as , the concentration about the line disappears and the distribution becomes truly uniform.

A shuffle of a deck of cards obtained by successively exchanging the cards in position 1, 2, ..., with cards in randomly chosen positions. For , the most frequent permutation is , where if is even and either or if is odd (Goldstine and Moews 2000). Amazingly, for cards, the identity permutation (i.e., the original state before the cards were shuffled) is the most likely (Goldstein and Moews 2000).

The self-describing sequence consisting of "blocks" of single and double 1s and 2s, where a "block" is a single digit or pair of digits that is different from the digit (or pair of digits) in the preceding block. To construct the sequence, start with the single digit 1 (the first "block"). Here, the single 1 means that block of length one follows the first block. Therefore, require that the next block is 2, giving the sequence 12.Now, the 2 means that the next (third) block will have length two, so append 11 and obtain the sequence 1211. We have added two 1s, so the fourth and fifth blocks have length one each, giving 12112 and then 121121. As a result of adding 21, we obtain 121121221. As a result of adding 221, we obtain 12112122122112, and so on, giving the sequence 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ... (OEIS A006928). The sequence after successive iterations is given by 1, 12, 1211, 121121, 121121221, ..., and the lengths..

In statistics, sampling is the selection and implementation of statistical observations in order to estimate properties of an underlying population. Sampling is a vital part of modern polling, market research, and manufacturing, and its proper use is vital in the functioning of modern economies. The portion of a population selected for analysis is known as a sample, and the number of members in the sample is called the sample size.The term "sampling" is also used in signal processing to refer to measurement of a signal at discrete times, usually with the intension of reconstructing the original signal. For infinite-precision sampling of a band-limited signal at the Nyquist frequency, the signal-to-noise ratio after samples is(1)(2)(3)where is the normalized correlation coefficient(4)For ,(5)The identical result is obtained for oversampling. For undersampling, the signal-to-noiseratio decreases (Thompson et al. 1986)...

Given a sequence , an -moving average is a new sequence defined from the by taking the arithmetic mean of subsequences of terms,(1)So the sequences giving -moving averages are(2)(3)and so on. The plot above shows the 2- (red), 4- (yellow), 6- (green), and 8- (blue) moving averages for a set of 100 data points.Moving averages are implemented in the Wolfram Language as MovingAverage[data, n].

A random matrix is a matrix of given type and size whoseentries consist of random numbers from some specified distribution.Random matrix theory is cited as one of the "modern tools" used in Catherine'sproof of an important result in prime number theory in the 2005 film Proof.For a real matrix with elements having a standard normal distribution, the expected number of real eigenvalues is given by(1)(2)where is a hypergeometric function and is a beta function (Edelman et al. 1994, Edelman and Kostlan 1994). has asymptotic behavior(3)Let be the probability that there are exactly real eigenvalues in the complex spectrum of the matrix. Edelman (1997) showed that(4)which is the smallest probability of all s. The entire probability function of the number of expected real eigenvalues in the spectrum of a Gaussian real random matrix was derived by Kanzieper and Akemann (2005) as(5)where(6)(7)In (6), the summation runs over all partitions..

Clean tile is a game investigated by Buffon (1777) in which players bet on the number of different tiles a thrown coin will partially cover on a floor that is regularly tiled. Buffon investigated the probabilities on a triangular grid, square grid, hexagonal grid, and grid composed of rhombi. Assume that the side length of the tile is greater than the coin diameter . Then, on a square grid, it is possible for a coin to land so that it partially covers 1, 2, 3, or 4 tiles. On a triangular grid, it can land on 1, 2, 3, 4, or 6 tiles. On a hexagonal grid, it can land on 1, 2, or 3 tiles.Special cases of this game give the Buffon-Laplace needle problem (for a square grid) and Buffon's needle problem (for infinite equally spaced parallel lines).As shown in the figure above, on a square grid with tile edge length , the probability that a coin of diameter will lie entirely on a single tile (indicated by yellow disks in the figure) is given by(1)since the shortening of the side..

Buffon's needle problem asks to find the probability that a needle of length will land on a line, given a floor with equally spaced parallel lines a distance apart. The problem was first posed by the French naturalist Buffon in 1733 (Buffon 1733, pp. 43-45), and reproduced with solution by Buffon in 1777 (Buffon 1777, pp. 100-104).Define the size parameter by(1)For a short needle (i.e., one shorter than the distance between two lines, so that ), the probability that the needle falls on a line is(2)(3)(4)(5)For , this therefore becomes(6)(OEIS A060294).For a long needle (i.e., one longer than the distance between two lines so that ), the probability that it intersects at least one line is the slightly more complicated expression(7)where (Uspensky 1937, pp. 252 and 258; Kunkel).Writing(8)then gives the plot illustrated above. The above can be derived by noting that(9)where(10)(11)are the probability functions for the..

The term domain has (at least) three different meanings in mathematics.The term domain is most commonly used to describe the set of values for which a function (map, transformation, etc.) is defined. For example, a function that is defined for real values has domain , and is sometimes said to be "a function over the reals." The set of values to which is sent by the function is then called the range.Unfortunately, the term range is sometimes used in probability theory to mean domain (Feller 1968, p. 200; Evans et al. 2000). To confuse matters even more, the term "range" is more commonly used in statistics to refer to a completely different quantity, known in this work as the statistical range. As if this wasn't confusing enough, Evans et al. (2000, p. 6) define a probability domain to be the range of the distribution function of a probability density function.The domain (in its usual established mathematical sense)..

The conditional intensity associated to a temporal point process is defined to be the expected infinitesimal rate at which events are expected to occur around time given the history of at times prior to time . Algebraically,provided the limit exists where here, is the history of over all times strictly prior to time .

If is a map (a.k.a. function, transformation, etc.) over a domain , then the range of , also called the image of under , is defined as the set of all values that can take as its argument varies over , i.e.,Note that among mathematicians, the word "image"is used more commonly than "range."The range is a subset of and does not have to be all of .Unfortunately, term "range" is often used to mean domain--its precise opposite--in probability theory, with Feller (1968, p. 200) and Evans et al. (2000, p. 5) calling the set of values that a variate can assume (i.e., the set of values that a probability density function is defined over) the "range", denoted by (Evans et al. 2000, p. 5).Even worse, statistics most commonly uses "range" to refer to the completely different statistical quantity as the difference between the largest and smallest order statistics. In this work, this form..

A Poisson process is a process satisfying the following properties: 1. The numbers of changes in nonoverlapping intervals are independent for all intervals. 2. The probability of exactly one change in a sufficiently small interval is , where is the probability of one change and is the number of trials. 3. The probability of two or more changes in a sufficiently small interval is essentially 0. In the limit of the number of trials becoming large, the resulting distribution iscalled a Poisson distribution.

The expected number of real zeros of a random polynomial of degree if the coefficients are independent and distributed normally is given by(1)(2)(Kac 1943, Edelman and Kostlan 1995). Another form of the equation is given by(3)(Kostlan 1993, Edelman and Kostlan 1995). The plots above show the integrand (left) and numerical values of (red curve in right plot) for small . The first few values are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955, ....As ,(4)where(5)(6)(OEIS A093601; top curve in right plot above).The initial term was derived by Kac (1943).

Given a subset of a larger set, the characteristic function , sometimes also called the indicator function, is the function defined to be identically one on , and is zero elsewhere. Characteristic functions are sometimes denoted using the so-called Iverson bracket, and can be useful descriptive devices since it is easier to say, for example, "the characteristic function of the primes" rather than repeating a given definition. A characteristic function is a special case of a simple function.The term characteristic function is used in a different way in probability, where it is denoted and is defined as the Fourier transform of the probability density function using Fourier transform parameters ,(1)(2)(3)(4)(5)where (sometimes also denoted ) is the th moment about 0 and (Abramowitz and Stegun 1972, p. 928; Morrison 1995).A statistical distribution is not uniquely specified by its moments, but is by its characteristic..

The randomization of a deck of cards by repeated interleaving. More generally, a shuffle is a rearrangement of the elements in an ordered list. Shuffling by exactly interleaving two halves of a deck is called a riffle shuffle. Shuffling by successively interchanging the cards in position 1, 2, ..., with cards in randomly chosen positions is known as an exchange shuffle. Normal shuffling leaves gaps of different lengths between the two layers of cards and so randomizes the order of the cards.Keller (1995) showed that roughly shuffles are needed just to randomize the bottom card.

Consider the probability that no two people out of a group of will have matching birthdays out of equally possible birthdays. Start with an arbitrary person's birthday, then note that the probability that the second person's birthday is different is , that the third person's birthday is different from the first two is , and so on, up through the th person. Explicitly,(1)(2)But this can be written in terms of factorials as(3)so the probability that two or more people out of a group of do have the same birthday is therefore(4)(5)In general, let denote the probability that a birthday is shared by exactly (and no more) people out of a group of people. Then the probability that a birthday is shared by or more people is given by(6)In general, can be computed using the recurrence relation(7)(Finch 1997). However, the time to compute this recursive function grows exponentially with and so rapidly becomes unwieldy.If 365-day years have been assumed, i.e.,..

There are several statistical quantities called means, e.g., harmonic mean, geometric mean, arithmetic-geometric mean, and root-mean-square. When applied to two elements and with , these means satisfy(1)The following table summarizes these means (again applied to two elements and with ), where is a complete elliptic integral of the first kind.meanvalueharmonic meangeometric meanarithmetic-geometric meanarithmetic meanroot-mean-squareThe quantity commonly referred to as "the" mean of a set of values is thearithmetic mean(2)also called the (unweighted) average. Notations for "the" mean of a set of values include macron notation or . The expectation value notation is sometimes also used. The mean of a list of data (i.e., the sample mean) is implemented as Mean[list].In general, a mean is a homogeneous function that has the property that a mean of a set of numbers satisfies(3)The term function centroid is..

The word net has several meanings in mathematics. It refers to a plane diagram in which the polyhedron edges of a polyhedron are shown, a point set satisfying certain uniformity of distribution conditions, and a topological generalization of a sequence.The net of a polyhedron is also known as a development, pattern, or planar net (Buekenhout and Parker 1998). The illustrations above show polyhedron nets for the cube and tetrahedron.In his classic Treatise on Measurement with the Compass and Ruler, Dürer(1525) made one of the first presentations of a net (Livio 2002, p. 138).The net of a polyhedron must in general also specify which edges are to be joined since there might be ambiguity as to which of several possible polyhedra a net might fold into. For simple symmetrical polyhedra, the folding procedure can only be done one way, so edges need not be labeled. However, for the net shown above, two different solids can be constructed from..

A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markov matrix, is matrix used to characterize transitions for a finite Markov chain, Elements of the matrix must be real numbers in the closed interval [0, 1].A completely independent type of stochastic matrix is defined as a square matrix with entries in a field such that the sum of elements in each column equals 1. There are two nonsingular stochastic matrices over (i.e., the integers mod 2),There are six nonsingular stochastic matrices over ,In fact, the set of all nonsingular stochastic matrices over a field forms a group under matrix multiplication. This group is called the stochastic group.The following tables give the number of distinct stochastic matrices (and distinct nonsingular stochastic matrices) over for small .stochastic matrices over 21, 4, 64, 4096, ...31, 9, 729, ...41, 16, 4096, ...stochastic..

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