In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, an excursion probability is the probability that a
stochastic process surpasses a given value in a fixed time period. It is the probability
:
Numerous approximation methods for the situation where ''u'' is large and f(''t'') is a
Gaussian process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...
have been proposed such as
Rice's formula
In probability theory, Rice's formula counts the average number of times an ergodic stationary process ''X''(''t'') per unit time crosses a fixed level ''u''. Adler and Taylor describe the result as "one of the most important results in the applica ...
.
First-excursion probabilities can be used to describe deflection to a critical point experienced by structures during "random loadings, such as earthquakes, strong gusts, hurricanes, etc."
References
Stochastic processes
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