## Quick Reference

Theory concerned with determining the probability that a system (with *n* components) is working. Let *x** _{j}*=1 if the

*j*th component is working and let

*x*

*=0 if it has failed. The vector*

_{j}**x**=(

*x*

_{1}

*x*

_{2}…

*x*

*) is called the state vector. The function*

_{n}*ϕ*(

**x**), which takes the value 1 when the system is working and 0 when it has failed, is called the structure function. For

*n*components in series,

*ϕ*(**x**)=min{*x*_{1}, *x*_{2},…, *x** _{n}*}=

*x*

_{1}×

*x*

_{2}×…×

*x*

*.*

_{n}For *n* components in parallel,

*ϕ*(**x**)=max{*x*_{1}, *x*_{2},…, *x** _{n}*}=1-(1-

*x*

_{1})(1-

*x*

_{2})…(1-

*x*

*).*

_{n}If *ϕ*(**x**)=1 then **x** is a path vector: it traces a set of connected working components. If failure of any of its working components results in system failure, the vector is a minimal path vector. Correspondingly, if *ϕ*(**x**)=0 then **x** is a cut vector and, if it is the case that repair of any of the failed components in **x** leads to the system working, then the vector is a minimal cut vector.

If *p** _{j}* denotes the probability that the

*j*th component continues to work during the next unit of time, then the probability that a structure consisting of

*n*components in series continues to work is

*p*

_{1}×

*p*

_{2}×···×

*p*

*with the corresponding probability for*

_{n}*n*components in parallel being 1-(1-

*p*

_{1})(1-

*p*

_{2})…(1-

*p*

*).*

_{n}**Reliability theory.** Complicated systems can always be subdivided into combinations of components arranged either in series or in parallel.

*Subjects:*
Probability and Statistics.

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