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Chi Distribution
In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. It is thus related to the chi-squared distribution by describing the distribution of the positive square roots of a variable obeying a chi-squared distribution. If Z_1, \ldots, Z_k are k independent, normally distributed random variables with mean 0 and standard deviation 1, then the statistic :Y = \sqrt is distributed according to the chi distribution. The chi distribution has one parameter, k, which specifies the number of degrees of freedom (i.e. the number of random variables Z_i). The most familiar examples are the Rayleigh distribution (chi distribution with two degrees of freedom) and the Maxwell–Boltzmann dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chi Distribution PDF
Chi or CHI may refer to: Greek *Chi (letter), the Greek letter (uppercase Χ, lowercase χ); Chinese * ''Chi'' (length) (尺), a traditional unit of length, about ⅓ meter *Chi (mythology) (螭), a dragon *Chi (surname) (池, pinyin: ''chí'') * ''Ch'i'' or ''qi'' (氣), "energy force" *Chinese language (ISO 639-2 code "chi") *Ji (surname), various surnames written Chi in Wade–Giles Arts and entertainment * ''Chi'' (2013 film), a Canadian documentary * ''Chi'' (2019 film), a Burmese drama *'' Chi: Chikyū no Undō ni Tsuite'', a manga series by Uoto *''The Chi'', an American drama series created by Lena Waithe for Showtime * Chi (''Chobits''), a character in ''Chobits'' media * Sailor Chi, a villain in the ''Sailor Moon'' manga *Chi, a character in ''Chi's Sweet Home'' media *"Chi", a song by Korn from '' Life Is Peachy'' Science and mathematics *Chi, the hyperbolic cosine integral *A symbol for electronegativity People *Chi Cheng (musician) (1970–2013), American musici ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment-generating Function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions. As its name implies, the moment-generating function can be used to compute a distribution’s moments: the ''n''th moment about 0 is the ''n''th derivative of the moment-generating function, evaluated at 0. In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases. The moment-generating func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nakagami Distribution
The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling spread \Omega>0. Characterization Its probability density function (pdf) is : f(x;\,m,\Omega) = \fracx^\exp\left(-\fracx^2\right), \forall x\geq 0. where (m\geq 1/2,\text\Omega>0) Its cumulative distribution function is : F(x;\,m,\Omega) = P\left(m, \fracx^2\right) where ''P'' is the regularized (lower) incomplete gamma function. Parametrization The parameters m and \Omega are : m = \frac , and : \Omega = \operatorname \left ^2 \right Parameter estimation An alternative way of fitting the distribution is to re-parametrize \Omega and ''m'' as ''σ'' = Ω/''m'' and ''m''. Given independent observations X_1=x_1,\ldots,X_n=x_n from the Nakagami distribution, the likelihood function is : L( \sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Gamma Distribution
The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in survival analysis (such as the exponential distribution, the Weibull distribution and the gamma distribution) are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data. Another example is the half-normal distribution. Characteristics The generalized gamma distribution has two shape parameters, d > 0 and p > 0, and a scale parameter, a > 0. For non-negative ''x'' from a generalized gamma distribution, the probability density function is : f(x; a, d, p) = \frac, where \Gamma(\cdot) denotes the gamma function. The cumulative distribution function is : F(x; a, d, p) = \frac , where \gamma(\cdot) denotes t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Normal Distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be ''k''-variate normally distributed if every linear combination of its ''k'' components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Definitions Notation and parameterization The multivariate normal distribution of a ''k''-dimensional random vector \mathbf = (X_1,\ldots,X_k)^ can be written in the following notation: : \mathbf\ \sim\ \mathcal(\boldsymbol\mu,\, \boldsymbol\Sigma), or to make it explicitly known that ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norm (mathematics)
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "\,\leq\," in the homogeneity axiom. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxwell Distribution
Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (other) * Maxwell baronets, in the Baronetage of Nova Scotia * Maxwell (footballer, born 1979), Brazilian forward * Maxwell (footballer, born 1981), Brazilian left-back * Maxwell (footballer, born 1986), Brazilian striker * Maxwell (footballer, born 1989), Brazilian left-back * Maxwell (footballer, born 1995), Brazilian forward * Maxwell (musician) (born 1973), American R&B and neo-soul singer * Maxwell (rapper) (born 1993), German rapper, member of rap band 187 Strassenbande * Maxwell Jacob Friedman (born 1997) AEW Professional wrestler * Maxwell Silva (born 1953), Sri Lankan Sinhala Catholic cleric, Auxiliary Bishop of Colombo Places United States * Maxwell, California * Maxwell, Indiana * Maxwell, Iowa * Maxwell, Nebraska * Maxwell, New Mexico * Maxwell, Texas * Maxwel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Half-normal Distribution
In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero. Properties Using the \sigma parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by : f_Y(y; \sigma) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if \sigma is near zero), obtained by setting \theta=\frac, the probability density function is given by : f_Y(y; \theta) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. The cumulative distribution function (CDF) is given by : F_Y(y; \sigma) = \int_0^y \frac\sqrt \, \exp \left( -\frac \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling's Approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqrt\left(\frac\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication Theorem
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a ''product''. The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''. Both numbers can be referred to as ''factors''. :a\times b = \underbrace_ For example, 4 multiplied by 3, often written as 3 \times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: :3 \times 4 = 4 + 4 + 4 = 12 Here, 3 (the ''multiplier'') and 4 (the ''multiplicand'') are the ''factors'', and 12 is the ''product''. One of the main properties of multiplication is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygamma Function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) = \psi(z) = \frac holds where is the digamma function and is the gamma function. They are holomorphic on \mathbb \backslash\mathbb_. At all the nonpositive integers these polygamma functions have a pole of order . The function is sometimes called the trigamma function. Integral representation When and , the polygamma function equals :\begin \psi^(z) &= (-1)^\int_0^\infty \frac\,\mathrmt \\ &= -\int_0^1 \frac(\ln t)^m\,\mathrmt\\ &= (-1)^m!\zeta(m+1,z) \end where \zeta(s,q) is the Hurwitz zeta function. This expresses the polygamma function as the Laplace transform of . It follows from Bernstein's theorem on monotone functions that, for and real and non-negative, is a completely monotone function. Setting in the above ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
