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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 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.
The random variable (Y/σ) 2 has a noncentral chi-squared distribution with 1 degree of freedom and noncentrality equal to (μ/σ) 2. The folded normal distribution can also be seen as the limit of the folded non-standardized t distribution as the degrees of freedom go to infinity.
A random vector X ∈ R p (a p×1 "column vector") has a multivariate normal distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ R p × p is a positive-definite matrix and the probability density function of X is
[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 .
Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a cumulative from mean table or 0.75490 from a cumulative table. To find a negative value such as -0.83, one could use a cumulative table for negative z-values [3] which yield a probability of 0.20327.
Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value μ = b and variance σ 2 = c 2. In this case, the Gaussian is of the form [1]
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 .