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. [1] Adler and Taylor describe the result as "one of the most important results in the applications of smooth stochastic processes." [2] The formula is often used in engineering. [3]
The formula was published by Stephen O. Rice in 1944, [4] having previously been discussed in his 1936 note entitled "Singing Transmission Lines." [5] [6]
Write Du for the number of times the ergodic stationary stochastic process x(t) takes the value u in a unit of time (i.e. t ∈ [0,1]). Then Rice's formula states that
where p(x,x') is the joint probability density of the x(t) and its mean-square derivative x'(t). [7]
If the process x(t) is a Gaussian process and u = 0 then the formula simplifies significantly to give [7] [8]
where ρ'' is the second derivative of the normalised autocorrelation of x(t) at 0.
Rice's formula can be used to approximate an excursion probability [9]
as for large values of u the probability that there is a level crossing is approximately the probability of reaching that level.
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