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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 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 [ 2 ] [ 3 ] f ( x ) = 1 2 π σ 2 e − ( x − μ ) 2 2 σ 2 . {\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2 ...
The Dirac comb of period 2 π, although not strictly a function, is a limiting form of many directional distributions. It is essentially a wrapped Dirac delta function. It represents a discrete probability distribution concentrated at 2 π n — a degenerate distribution — but the notation treats it as if it were a continuous distribution.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
The characteristic function + = ((+)) of the sum of two independent random variables X and Y is just the product of the two separate characteristic functions: = (), = ()
Diagram showing the cumulative distribution function for the normal distribution with mean (μ) 0 and variance (σ 2) 1. These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score z corresponds numerically to (1 − (1 − Φ μ,σ 2 (z)) · 2).
If one of the principal curvatures is zero: κ 1 κ 2 = 0, the Gaussian curvature is zero and the surface is said to have a parabolic point. Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a ...
If Y has a half-normal distribution, then (Y/σ) 2 has a chi square distribution with 1 degree of freedom, i.e. Y/σ has a chi distribution with 1 degree of freedom. The half-normal distribution is a special case of the generalized gamma distribution with d = 1, p = 2, a = 2 σ {\displaystyle {\sqrt {2}}\sigma } .