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The probability density, cumulative distribution, and inverse cumulative distribution of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing [41] (Matlab code). In the following sections we look at some special cases.
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
In statistics, the Q-function is the tail distribution function of the standard normal distribution. [ 1 ] [ 2 ] In other words, Q ( x ) {\displaystyle Q(x)} is the probability that a normal (Gaussian) random variable will obtain a value larger than x {\displaystyle x} standard deviations.
The Dirac delta function, although not strictly a probability distribution, is a limiting form of many continuous probability functions. It represents a discrete probability distribution concentrated at 0 — a degenerate distribution — it is a Distribution (mathematics) in the generalized function sense; but the notation treats it as if it ...
In this example, the ratio (probability of living during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour −1). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 2 hour −1.
An arbitrary function φ : R n → C is the characteristic function of some random variable if and only if φ is positive definite, continuous at the origin, and if φ(0) = 1. Khinchine’s criterion. A complex-valued, absolutely continuous function φ, with φ(0) = 1, is a characteristic function if and only if it admits the representation
The convolution of a function with a Gaussian is also known as a Weierstrass transform. A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator. The molecular orbitals used in computational chemistry can be linear combinations of Gaussian functions called Gaussian orbitals (see also basis set (chemistry)).
Quantile functions are used in both statistical applications and Monte Carlo methods. The quantile function is one way of prescribing a probability distribution, and it is an alternative to the probability density function (pdf) or probability mass function, the cumulative distribution function (cdf) and the characteristic function.