Probability Integral

Probability Integral


the name of several mutually related special functions. The integral

is called the Gaussian probability integral. For a random quantity X having a normal distribution with an expected value 0 and variance σ2, the probability of the inequality ǀXǀ ≤ x is equal to Φ(x/≤. In addition to this, the name “probability integral” is used for the integrals

The latter function is usually designated erf (x) (from “error function”).


Bol’shev, L. N., and N.V. Smirnov. Tablitsy matematicheskoi statistiki. Moscow, 1965.
References in periodicals archive ?
As a matter of fact, the surface deformations along a pipeline could be predicted in advance through the probability integral method, if subsidence prediction parameters of a mine are given [3-5].
The surface subsidence along the pipeline obtained by the probability integral method is substituted in (1).
Each one of these evaluations includes the computation of 2*6+1=13 conditional probabilities of failure, needed by the sparse grid integration to compute the total probability integral (Equations 2 and 30).
The transformation is called the probability integral transformation ([3], [4, pages 203-204]).
Quesenberry, "Probability integral transformations," in Encyclopedia of Statistical Sciences, S.
In order to test the performance of this improved algorithm, both simulated image pair named simulated data with known displacements by probability integral method for each pixel and real ground deformation caused by mining in the Daliuta mining area named real ground subsidence data are used in this paper.
The simulated ground subsidence is predicted by probability integral method (shown in Figure 2).
It follows from the probability integral transform that the random variables [mathematical expression not reproducible], are all uniformly distributed on (0,1).
where [PHI] is probability integral and [M.sub.Ek]--characteristic value of acting moment.
where [PHI] is probability integral; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--standard deviation of load carrying capacity.
where p is a specified value for the probability integral.
Initial chapters describe theoretical aspects, including probability integrals and inequalities.
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