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Complex Normal Distribution
In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''location'' parameter ''μ'', ''covariance'' matrix \Gamma, and the ''relation'' matrix C. The standard complex normal is the univariate distribution with \mu = 0, \Gamma=1, and C=0. An important subclass of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean: \mu = 0 and C=0 . This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature. Definitions Complex standard normal random variable The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable Z whose real and imaginary parts are independent normally distributed random v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Location Parameter
In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x_0, which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formally defined in one of the following equivalent ways: * either as having a probability density function or probability mass function f(x - x_0); or * having a cumulative distribution function F(x - x_0); or * being defined as resulting from the random variable transformation x_0 + X, where X is a random variable with a certain, possibly unknown, distribution. See also . A direct example of a location parameter is the parameter \mu of the normal distribution. To see this, note that the probability density function f(x , \mu, \sigma) of a normal distribution \mathcal(\mu,\sigma^2) can have the parameter \mu factored out and be written as: : g(x' = x - \mu , \sigma) = \frac \exp\left(-\f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Random Variable
In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the #Cumulative distribution function, distribution of one complex random variable may be interpreted as the Joint probability distribution, joint distribution of two real random variables. Some concepts of real random variables have a straightforward generalization to complex random variables—e.g., the definition of the #Expectation, mean of a complex random variable. Other concepts are unique to complex random variables. Applications of complex random variables are found in digital signal processing, quadrature amplitude modulation and information theory. Definition A complex random variable Z on the probability space (\Omega,\mathcal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wishart Distribution
In statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart (statistician), John Wishart, who first formulated the distribution in 1928. Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with Random matrix#Gaussian ensembles, GOE, GUE, GSE). It is a family of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix (mathematics), matrix-valued random variables). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. In Bayesian inference, Bayesian statistics, the Wishart distribution is the conjugate prior of the matrix inverse, inverse covariance matrix, covariance-matrix of a multivariate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Chi-squared Distribution
In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic function of a multivariate normal distribution, multinormal variable (normal vector), or a linear combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independent noncentral chi-squared distribution, noncentral chi-square variables and a normal distribution, normal variable. There are several other such generalizations for which the same term is sometimes used; some of them are special cases of the family discussed here, for example the gamma distribution. Definition The generalized chi-squared variable may be described in multiple ways. One is to write it as a weighted sum of independent Noncentral chi-squared distribution, noncentral chi-square variables ^2 and a standard normal variable z: :\tilde(\boldsymbol, \boldsymbol, \boldsymbol,s,m)=\sum_i w_i ^2 (k_i,\la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In probability theory and 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 is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). 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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Normal Ratio Distribution
A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables ''X'' and ''Y'', the distribution of the random variable ''Z'' that is formed as the ratio ''Z'' = ''X''/''Y'' is a ''ratio distribution''. An example is the Cauchy distribution (also called the ''normal ratio distribution''), which comes about as the ratio of two normally distributed variables with zero mean. Two other distributions often used in test-statistics are also ratio distributions: the ''t''-distribution arises from a Gaussian random variable divided by an independent chi-distributed random variable, while the ''F''-distribution originates from the ratio of two independent chi-squared distributed random variables. More general ratio distributions have been considered in the literature. Often the ratio distributions are heavy-t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Wishart Distribution
In statistics, the complex Wishart distribution is a complex version of the Wishart distribution. It is the distribution of n times the sample Hermitian covariance matrix of n zero-mean independent Gaussian random variables. It has support for p\times p Hermitian positive definite matrices. The complex Wishart distribution is the density of a complex-valued sample covariance matrix. Let : S_ = \sum_^n G_iG_i^H where each G_i is an independent column ''p''-vector of random complex Gaussian zero-mean samples and (.)^H is an Hermitian (complex conjugate) transpose. If the covariance of ''G'' is \mathbb G^H= M then : S \sim n\mathcal(M,n,p) where \mathcal(M,n,p) is the complex central Wishart distribution with ''n'' degrees of freedom and mean value, or scale matrix, ''M''. : f_S(\mathbf) = \frac , \;\;\; n\ge p, \;\;\; \left, \mathbf\ > 0 where : \mathcal \widetilde_p^ (n) = \pi^ \prod_^p \Gamma (n-j+1) is the complex multivariate Gamma function. Using the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Chi-squared Distribution
In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic function of a multivariate normal distribution, multinormal variable (normal vector), or a linear combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independent noncentral chi-squared distribution, noncentral chi-square variables and a normal distribution, normal variable. There are several other such generalizations for which the same term is sometimes used; some of them are special cases of the family discussed here, for example the gamma distribution. Definition The generalized chi-squared variable may be described in multiple ways. One is to write it as a weighted sum of independent Noncentral chi-squared distribution, noncentral chi-square variables ^2 and a standard normal variable z: :\tilde(\boldsymbol, \boldsymbol, \boldsymbol,s,m)=\sum_i w_i ^2 (k_i,\la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, a and b, which are the minimum and maximum values. The interval can either be closed (i.e. ,b/math>) or open (i.e. (a,b)). Therefore, the distribution is often abbreviated U(a,b), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable X under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is f(x) = \begin \dfrac & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Distribution
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh Distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribution is named after Lord Rayleigh (). A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, which is infrequent, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
