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In statistics, a Gaussian random field (GRF) is a random field involving Gaussian probability density functions of the variables. A one-dimensional GRF is also called a Gaussian process . An important special case of a GRF is the Gaussian free field .
In probability theory particularly in the Malliavin calculus, a Gaussian probability space is a probability space together with a Hilbert space of mean zero, real-valued Gaussian random variables. Important examples include the classical or abstract Wiener space with some suitable collection of Gaussian random variables.
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.
An example found by Marcus and Shepp [18]: 387 is a random lacunary Fourier series = = ( + ), where ,,,, … are independent random variables with standard normal distribution; frequencies < < < … are a fast growing sequence; and coefficients > satisfy <.
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution , (,,) has the form: (,,) = ([() ()]) / | | / | | /where denotes trace and M is n × p, U is n × n and V is p × p, and the density is understood as the probability density function with respect to the standard Lebesgue measure in , i.e.: the measure corresponding to integration ...
A typical example of a circular symmetric complex random variable is the complex Gaussian random variable with zero mean and zero pseudo-covariance matrix. A complex random variable Z {\displaystyle Z} is circularly symmetric if, for any deterministic ϕ ∈ [ − π , π ] {\displaystyle \phi \in [-\pi ,\pi ]} , the distribution of e i ϕ Z ...
[1] [2] In other words, () is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations. Equivalently, () is the probability that a standard normal random variable takes a value larger than .
The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable whose real and imaginary parts are independent normally distributed random variables with mean zero and variance /. [3]: p. 494 [4]: pp. 501 Formally,