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Raised Cosine Distribution
In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval mu-s,\mu+s/math>. The probability density function (PDF) is :f(x;\mu,s)=\frac \left +\cos\left(\frac\,\pi\right)\right,=\frac\operatorname\left(\frac\,\pi\right) \text \mu-s\le x\le\mu+s and zero otherwise. The cumulative distribution function (CDF) is :F(x;\mu,s)=\frac\left +\frac + \frac \sin\left(\frac \, \pi \right) \right/math> for \mu-s \le x \le \mu+s and zero for x\mu+s. The 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 function, the odd moments are zero. The even moments are given by: : \begin \operatorname E(x^) & = \frac\int_^1 +\cos(x\pi)^\,dx = \int_^1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raised Cos Pdf Mod
''Raised'' is the third studio album by American country music, country artist Hailey Whitters. It was released on March 18, 2022, via a partnership between Big Loud and her own imprint, Pigasus. Background Heavily inspired by her Midwestern upbringing in Iowa, Whitters has writing credits on all the tracks on ''Raised'', with the exception of "Everybody Oughta", and the album's instrumental intro track ("Ad Astra Per Alas Porci") and its reprise as the album closer. She co-produced the 17-song project with Jake Gear. Whitters felt like her previous album, ''The Dream'', drew largely from her experience in Nashville, Tennessee, but wanted to take things back to her roots with ''Raised'' and called it a "celebration of the heartland". She described the song "Heartland" from her previous album as being the "mustard seed" that led to her reminiscing on her hometown and childhood memories. On ''Raised'', Whitters said "these songs are airy, breathe, and organically feel good because t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raised Cos Cdf Mod
''Raised'' is the third studio album by American country artist Hailey Whitters. It was released on March 18, 2022, via a partnership between Big Loud and her own imprint, Pigasus. Background Heavily inspired by her Midwestern upbringing in Iowa, Whitters has writing credits on all the tracks on ''Raised'', with the exception of "Everybody Oughta", and the album's instrumental intro track ("Ad Astra Per Alas Porci") and its reprise as the album closer. She co-produced the 17-song project with Jake Gear. Whitters felt like her previous album, ''The Dream'', drew largely from her experience in Nashville, Tennessee, but wanted to take things back to her roots with ''Raised'' and called it a "celebration of the heartland". She described the song "Heartland" from her previous album as being the "mustard seed" that led to her reminiscing on her hometown and childhood memories. On ''Raised'', Whitters said "these songs are airy, breathe, and organically feel good because there's so much ... [...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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), 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 (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "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. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical description of a Randomness, random phenomenon in terms of its sample space and the Probability, probabilities of Event (probability theory), 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 fair coin, the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. Introduction A prob ... [...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 of elements that 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 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. The notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), 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 equal 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. More precisely, the PDF is used to specify the probability of the random variable falling ''within ... [...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. 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 systematically in terms of the moments of random variables. Significance of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Functions
In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is even if ''n'' is an even integer, and it is odd if ''n'' is an odd integer. Even functions are those real functions whose graph is self-symmetric with respect to the and odd functions are those whose graph is self-symmetric with respect to the origin. If the domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function. Early history The concept of even and odd functions appears to date back to the early 18th century, with Leonard Euler playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using La ... [...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 orthogonal polynomials, 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 succe ... [...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 or hanning. 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[n] = L\cdot w_0\left(\tfrac (n-N/2)\right) &= \tfrac \left[1 - \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. It is also known as the raised cosine window, Hann filter, von Hann window, Hanning 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 der ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
