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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.
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.
The generalized normal distribution (GND) or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To distinguish the two families, they are referred to below as "symmetric" and "asymmetric"; however ...
[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 .
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 <.
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 ...
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,
Since a non-central chi-squared variable is a sum of squares of normal variables with different means, the generalized chi-square variable is also defined as a sum of squares of independent normal variables, plus an independent normal variable: that is, a quadratic in normal variables.