reliability

The reliability function, R, is defined to be the probability that the component will not fail before time t. Thus

$$begin{aligned} R(t) &=1-P[text { component fails before time } t] \ &=1-int_{0}^{t} f(x) d x\ &=1-F(t) end{aligned}$$

The hazard rate $rho$ :

$$rho (t) = dfrac{f(t)}{R(t)}$$

Theorem:

$$R(t) = e^{-int_0^t rho(x)dx}$$

Reliability of Series and Parallel Systems:

The reliability of a series system wih k components:

$$R_{s}(t)=prod_{i=1}^{k} R_{i}(t)$$

The reliability of a parallel system is given by:

$$begin{aligned} R_{p}(t) &=1-P[text { all components fail before } t] \ &=1-prod_{i=1}^{k}left(1-R_{i}(t)right) end{aligned}$$