hesela.devWeibull distribution
A Weibull distribution is a continuous probability model whose two-parameter form uses positive shape and scale parameters to describe nonnegative event times.
With shape k > 0 and scale eta > 0, S(t) = exp(-(t/eta)^k) for t >= 0, and h(t) = (k/eta)(t/eta)^(k-1) for t > 0. Eta uses the time unit of the modeled variable; k is dimensionless. Shape below, equal to or above one gives decreasing, constant or increasing hazard. The k = 1 case is exponential. [1]
At t = eta, the cumulative event probability is 1 - exp(-1), approximately 63.2%, regardless of shape. This follows from the CDF; eta is not generally the mean or median. A single two-parameter Weibull has monotone hazard, so cannot by itself represent a decrease followed by an increase. These are model properties, not measured drive behavior. [1]
Specify the event and clock before interpreting a fitted shape. FAST 2007 fitted HPC1 intervals between cluster disk replacements, not individual disk lifetimes. A marginal interval fit does not establish independence between consecutive events; the paper separately examined correlation. Do not transfer those fitted parameters into a device-aging model. [2]
Conditioning on survival to age a gives P(T <= a + u | T > a) = 1 - S(a + u)/S(a), for a >= 0 and u >= 0. This model deduction differs from the unconditional probability by age a + u; the latter includes events before a. For Weibull k = 1, the conditional probability is independent of age. [1]
Relations
- related
- Hazard rate, Survival function, Interarrival time, Annualized failure rate (AFR), Mean residual life