Reliability Statistics and Probability
Reliability Statistics and Probability
This domain tests distribution selection, parameter interpretation, and life measures such as B10 life, with emphasis on Weibull and exponential models.
Core rules
- Exponential: constant hazard λ. Weibull reliability R(t)=exp[-(t/η)^β]; β<1 is decreasing hazard, β=1 is exponential, β>1 is increasing hazard.
- η is the scale parameter: 63.2% of units fail by t=η for any β. β is shape, not a failure rate.
- Hazard h(t)=f(t)/R(t); Weibull hazard is (β/η)(t/η)^(β-1).
- Lognormal fits repair times and some fatigue data; binomial models pass/fail counts; Poisson models failures in time.
- B_p life: t=η[-ln(1-p)]^(1/β).
Worked example
A Weibull model has β=2.5 and η=10,000 h. R(2,000)=exp[-(0.2)^2.5]. Since (0.2)^2.5=0.01789, R=0.9823. B10 life=10,000[-ln(0.9)]^(1/2.5)=10,000(0.10536)^0.4=4,065 h. Interpretation: 10% of units fail by 4,065 h; B10 is not the time at which the failure rate equals 10%.
Traps
- Reading β=2.5 as a failure rate.
- Using the exponential model for clearly increasing-hazard data.
- Ignoring suspensions or censoring in maximum-likelihood estimation.
- Confusing B10 with mean life or with 10% of MTBF.