Weibull Distribution

The Weibull shape parameter (beta, also written as b or k) that determines the form of the failure distribution and the nature of the failure rate over time. Beta < 1 indicates decreasing failure rate (infant mortality); beta = 1 indicates constant failure rate (random failures, equivalent to exponential distribution); beta > 1 indicates increasing failure rate (wear-out). The value of beta reveals the physics of the dominant failure mechanism.

The Weibull scale parameter (eta, also called characteristic life) that represents the time by which 63.2% of the population is expected to have failed. Eta defines the horizontal scale of the Weibull distribution and is always expressed in the same units as the failure time data (hours, cycles, km). At t = eta, the reliability R(t) = e^(-1) = 0.368, regardless of the shape parameter.

The time by which a specified percentage of a population is expected to have failed, derived from the Weibull distribution parameters. B10 life is the time by which 10% of the population has failed (90% survival); B50 is median life (50% survival); B90 is the time by which 90% have failed. B-life values are used to set preventive replacement intervals and warranty periods.

A graph of the hazard rate versus time that describes the typical life cycle of a population of items, characterized by three distinct phases: (1) Infant Mortality — a decreasing failure rate period caused by manufacturing defects, installation errors, or design weaknesses; (2) Useful Life — a period of approximately constant, low failure rate representing random failures; (3) Wear-Out — an increasing failure rate period caused by age-related degradation mechanisms.

The instantaneous rate of failure at time t, given that the item has survived to time t. Hazard rate (also called the instantaneous failure rate or hazard function) describes how the risk of failure changes over the item's life. Unlike failure rate (which is an average), hazard rate is a time-dependent function that captures the changing nature of failure risk as equipment ages.

The probability that an item will perform its required function under stated conditions for a specified period of time. Reliability is a quantitative measure expressed as a probability (0 to 1) or percentage. It is a function of the failure rate and time: for constant failure rate systems, R(t) = e^(-lambda*t). Higher reliability means lower probability of failure over the same period.