Chapter 1

E 1.1. The distribution governing the life duration T of a system is without atom, having support ? ₊. Show that the probability, whereby a failure occurs at a time which is a fully positive natural number or rational value, is zero.

E 1.2. The distribution Q of the life duration T of a system has a distribution function F, given by the relationship:

where a > 0, ? > 0.

1. Calculate Q( a).

2. Calculate the MTTF and the variance of T.

3. Study the characteristics of the magnitudes that are calculated, when a ? 0+.

E 1.3. Let R(t) be the reliability function of a component/system, and suppose that there exist a positive number a, such that lim _{t ??} e ^atR( t) = 0. Show that the MTTF exists and

E 1.4. The lifetime of a system is the r.v T with c.d.f F on the half-real axis x ? 0. Its mission duration is a v.a. ? of c.d.f G again over x ? 0. Calculate the probability whereby the system will successfully accomplish its mission.

E 1.5. Let there be a component under continuous working. When it breaks down, it is replaced instantaneously by a new component. If the p.d.f for the survival time of the components is f( t) = ?exp{- ? t}, t