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A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. ... with zero mean and unit variance) ...
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,
To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.
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 ...
is a multivariate Gaussian random variable. [1] ... As such, almost all sample paths of a mean-zero Gaussian process with positive definite kernel ...
The normal distribution, also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the central limit theorem: every variable that can be modelled as a sum of many small independent, identically distributed variables with finite mean and variance is approximately normal. The normal-exponential-gamma distribution
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
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. [1] [2]