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Raised-cosine Distribution
In probability theory and statistics, the raised cosine distribution is a continuous probability distribution support (mathematics), supported on the interval [\mu-s,\mu+s]. The probability density function (PDF) is :f(x;\mu,s)=\frac \left[1+\cos\left(\frac\,\pi\right)\right]\,=\frac\operatorname\left(\frac\,\pi\right)\, for \mu-s \le x \le \mu+s and zero otherwise. The cumulative distribution function (CDF) is :F(x;\mu,s)=\frac\left[1+\frac + \frac \sin\left(\frac \, \pi \right) \right] for \mu-s \le x \le \mu+s and zero for x\mu+s. The moment (mathematics), moments of the raised cosine distribution are somewhat complicated in the general case, but are considerably simplified for the standard raised cosine distribution. The standard raised cosine distribution is just the raised cosine distribution with \mu=0 and s=1. Because the standard raised cosine distribution is an Even and odd functions, even function, the odd moments are zero. The even moments are given by: : \begin \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raised Cos Pdf Mod
Raise may refer to: Music *'' Raise!'', the name of a 1981 album by Earth, Wind, and Fire * '' Raise'' (album), the name of a 1991 album by Swervedriver Place names * Raise, Cumbria, England * Raise (Lake District), the name of the 12th highest mountain in the Lake District on the north-west coast of England Computing * , a PL/SQL error-handling command * , an exception handling command in the Python programming language Other *To bring up, see parenting * Raise, a term used in poker * Raise (mining), a vertical or inclined underground passageway in a mine *An increase in salary *The process of using a leavening agent in baking and brewing *A term used in magic (supernatural), meaning: to summon or conjure * Raise.com, an e-commerce gift card marketplace See also * RAISE (other) * Raising (other) * Relief Relief is a sculptural method in which the sculpted pieces are bonded to a solid background of the same material. The term '' relief'' is from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raised Cos Cdf Mod
Raise may refer to: Music *''Raise!'', the name of a 1981 album by Earth, Wind, and Fire * '' Raise'' (album), the name of a 1991 album by Swervedriver Place names * Raise, Cumbria, England *Raise (Lake District), the name of the 12th highest mountain in the Lake District on the north-west coast of England Computing * , a PL/SQL error-handling command * , an exception handling command in the Python programming language Other *To bring up, see parenting *Raise, a term used in poker *Raise (mining), a vertical or inclined underground passageway in a mine *An increase in salary *The process of using a leavening agent in baking and brewing *A term used in magic (supernatural), meaning: to summon or conjure *Raise.com, an e-commerce gift card marketplace See also * RAISE (other) * Raising (other) * Relief Relief is a sculptural method in which the sculpted pieces are bonded to a solid background of the same material. The term ''relief'' is from the La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is an even function if ''n'' is an even integer, and it is an odd function if ''n'' is an odd integer. Definition and examples Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on. The given e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Hypergeometric Function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of successive coefficients is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hann Function
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by: : w_0(x) \triangleq \left\. For digital signal processing, the function is sampled symmetrically (with spacing L/N and amplitude 1): : \left . \begin w = L\cdot w_0\left(\tfrac (n-N/2)\right) &= \tfrac \left - \cos \left ( \tfrac \right) \right\ &= \sin^2 \left ( \tfrac \right) \end \right \},\quad 0 \leq n \leq N, which is a sequence of N+1 samples, and N can be even or odd. (see ) It is also known as the raised cosine window, Hann filter, von Hann window, etc. Fourier transform The Fourier transform of w_0(x) is given by: :W_0(f) = \frac\frac = \frac Discrete transforms The Discrete-time Fourier transform (DTFT) of the N+1 length, time-shifted sequence is defined by a Fourier series, which also has a 3-term equivalent that is derived similarly to the Four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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