By Julian Schwinger; et al

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S/ D k/ D e t . t/ D t, which explains why is called the ‘rate’ of the process. To determine if an arbitrary counting process is actually a Poisson process, we must show that conditions (i)–(iii) are satisfied. Conditions (i) and (ii) can usually be verified from our knowledge of the process. However, it is not at all clear how we would determine that condition (iii) is satisfied, and for this reason we need an alternative definition of a Poisson process. 5. t/ D 1. t/. h/; where the final two equations follow from (ii), (iii), and (iv).

Let the n components fail at times T1Wn Ä T2Wn Ä Ä TnWn . , for j D 1; : : : ; n, Tj Wn W ˛j 1 failed ! 1/ ˛j ! 2/ ˛j ! ::: 1) failed (j ! tkWn j Dk kD1 Ä tnWn . We call this distribution the FMVE. where 0 D t0Wn Ä t1Wn Ä The formulae become more complicated as the number of components becomes larger than two. However, if we assume these different components are the same versions with equal parameters, then the reliability functions have simpler forms. If k components failed, the conditional distribution of the surviving components will have exponential distributions with parameters ˛kC1 , for k D 0; 1; : : : ; n 1.

The recurrent state j is called positive recurrent if jj < 1 and it is null recurrent if jj D 1. It can be shown that for a finite-state Markov chain, all recurrent states are positive recurrent. Tij / is the mean first passage time, where Tij D minfn 0 W Xn D j jX0 D i g. 3. Positive recurrent states that are aperiodic are called ergodic states. A Markov chain is said to be ergodic if all its states are ergodic states. 13. Consider Äa two-state Markov chain S D f1; 2g with transition 1=3 2=3 probability matrix P D .

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