industrial//reliability//Weibull distribution

The Weibull distribution is a two-parameter probability distribution for the time to failure of a component, and it is the working model of reliability engineering because one of its parameters says, by itself, whether units fail young, at random or from wear, which decides whether replacing them early makes sense. It arises naturally as the life of a chain that breaks at its weakest link, which is why it fits bearings, insulation, fatigue cracks and many other things that fail where the first of many flaws gives way. Its reliability function and hazard function are

The Weibull distribution is a two-parameter probability distribution for the time to failure of a component, and it is the working model of reliability engineering because one of its parameters says, by itself, whether units fail young, at random or from wear, which decides whether replacing them early makes sense. It arises naturally as the life of a chain that breaks at its weakest link, which is why it fits bearings, insulation, fatigue cracks and many other things that fail where the first of many flaws gives way. Its reliability function and hazard function are

R(t)=e−(t/η)β,h(t)=βη(tη)β−1.R(t)=e^{-(t/\eta)^{\beta}},\qquad h(t)=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}.R(t)=e−(t/η)β,h(t)=ηβ​(ηt​)β−1.

The characteristic life η\etaη sets the time scale: at t=ηt=\etat=η, 63.2 % of the population has failed, whatever the shape. The shape parameter β\betaβ tells the story. With β<1\beta<1β<1 the hazard falls with age: infant mortality, defects of manufacture or assembly. With β=1\beta=1β=1 it is constant, failures are random (shocks, voltage surges) and the distribution is the memoryless exponential distribution. With β>1\beta>1β>1 it rises: wear, fatigue, corrosion; around β=3.5\beta=3.5β=3.5 the life looks roughly bell-shaped.

Only β>1\beta>1β>1 justifies replacing a part before it fails. With a constant or falling hazard, a new part is no safer than the old one, and calendar replacement spends spares or brings in fresh defects (preventive maintenance).

Fitting needs failure times, and a plant has few failures and many units still running. Those survivors are information (their life is known to exceed their current age), and they enter the fit as right-censored observations; discarding them is the classic error (censored data). With scipy.stats.weibull_min.fit on a CensoredData object, five failures between 410 and 1,130 hours and fifteen units still running at 1,200 hours give β≈2.2\beta\approx2.2β≈2.2 and η≈2,100\eta\approx2{,}100η≈2,100 hours.

The traditional check is the Weibull plot, failure fractions on axes that turn a Weibull into a straight line whose slope is β\betaβ; a bend in it means two failure modes mixed, each with its own β\betaβ, and they should be separated before deciding anything.

It is one member of the family of life distributions in survival analysis, and an extreme-value distribution in its own right, chosen for the tails where the normal distribution is optimistic.