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 ...

, Rice's formula counts the average number of times an

ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...

stationary process In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...

''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 applications of smooth stochastic processes." The formula is often used in engineering.

History

The formula was published by

Stephen O. Rice Stephen Oswald Rice (November 29, 1907 – November 18, 1986) was a pioneer in the related fields of information theory, communications theory, and telecommunications. Biography Rice was born in Shedds, Oregon (later renamed Shedd). He received a ...

in 1944, having previously been discussed in his 1936 note entitled "Singing Transmission Lines."

Formula

Write ''D''_''u'' for the number of times the ergodic stationary stochastic process ''x''(''t'') takes the value ''u'' in a unit of time (i.e. ''t'' ∈ ,1. Then Rice's formula states that ::

\mathbb E(D_u) = \int_^\infty , x', p(u,x') \, \mathrmx'

where ''p''(''x'',''x''') is the joint probability density of the ''x''(''t'') and its mean-square derivative ''x(''t''). If the process ''x''(''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. e ...

and ''u'' = 0 then the formula simplifies significantly to give ::

\mathbb E(D_0) = \frac \sqrt

where ''ρ'''' is the second derivative of the normalised

autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

of ''x''(''t'') at 0.

Uses

Rice's formula can be used to approximate an

excursion probability In probability theory, an excursion probability is the probability that a stochastic process surpasses a given value in a fixed time period. It is the probability :\mathbb P \left\. Numerous approximation methods for the situation where ''u'' is la ...

\mathbb P \left\

as for large values of ''u'' the probability that there is a level crossing is approximately the probability of reaching that level.

References

Ergodic theory {{probability-stub