(^) x(t)
t
y 1 y (^2) y 3 y 4
t 1 t 2 t 3 t 4
E()t
Figure 8.2: Illustration of a realisation of a Poisson spike process.
If ( ) t is the intensity of the Poisson spike process ( )X t then it can be shown that the number
of realisations of the process ( )X t above the level (Et), out-crossings, is also a Poisson
process with the intensity:
^4 ()ttFt()(1 Y( ()))E (8.4)
Assuming that failure occurs the first time the process X()t outcrosses the level E()t the
probability of failure can be assessed from:
( ) 1 exp( 0 ( )(1 ( ( ))) )
t
Ptf 3 FYE3 d 3 (8.5)
This result is important as it is used to approximate more complex situations, as shall be
shown.
Normal Processes
A stochastic process X()t
2
is said to be Normal or equivalently Gaussian if any set of random
variables ( ),X tii 1, 0 n is jointly Normal distributed.
For such processes it can be shown that E^ ( ( ))t , the mean number of out-crossings per time
unit or Mean Out-crossing Rate (MOR) above the level E()t can be determined by the so-
called Rice’s formula, see e.g. Lin (1984):
,
()
( ( )) XX( , )( )
ttxx
EE - E Edx
(^)
(8.6)
where ,X E are the time derivative of the stochastic process X()t and the level E()t
respectively and -XX,(,)E x is the joint Normal probability density function of X and X. A
realisation of a Normal process is illustrated in Figure 8.3. Rice’s formula may be interpreted
as the probability that the random process ( )X t has a realisation exactly on the level ( )E t and
that the velocity of the processX( )t is higher than the velocity of the level ( )Et.