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45:. 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.
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Barnett, J. T. (2001). "Zero-Crossings of Random
Processes with Application to Estimation Detection". In Marvasti, Farokh A. (ed.).
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482:
Rychlik, I. (2000). "On Some
Reliability Applications of Rice's Formula for the Intensity of Level Crossings".
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Borovkov, K.; Last, G. (2012). "On Rice's formula for stationary multivariate piecewise smooth processes".
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the probability that there is a level crossing is approximately the probability of reaching that level.
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in 1944, having previously been discussed in his 1936 note entitled "Singing
Transmission Lines."
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202:{\displaystyle \mathbb {E} (D_{u})=\int _{-\infty }^{\infty }|x'|p(u,x')\,\mathrm {d} x'}
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334:{\displaystyle \mathbb {E} (D_{0})={\frac {1}{\pi }}{\sqrt {-\rho ''(0)}}}
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Adler, Robert J.; Taylor, Jonathan E. (2007). "Excursion
Probabilities".
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449:{\displaystyle \mathbb {P} \left\{\sup _{t\in }X(t)\geq u\right\}}
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for the number of times the ergodic stationary stochastic process
261: = 0 then the formula simplifies significantly to give
706:"The Expected Number of Zeros of a Stationary Gaussian Process"
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Stochastic
Calculus: Applications in Science and Engineering
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741:. Springer Monographs in Mathematics. pp. 75–76.
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612:Rainal, A. J. (1988). "Origin of Rice's formula".
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517:Adler, Robert J.; Taylor, Jonathan E. (2007).
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41:) per unit time crosses a fixed level
27:counts the average number of times an
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86:in a unit of time (i.e.
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