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Weibull Distribution

The Weibull distribution can be used to model many different failure distributions. Given a shape parameter (β) and characteristic life (η) the reliability can be determined at a specific point in time (t). The two-parameter Weibull distribution probability density function, reliability function and hazard rate are given by:
Weibull Distribution PDF Equation Probability Density Function
Weibull Distribution Reliability Function Reliability Function
Weibull Distribution Hazard Rate Hazard Rate

The characteristic life (η) is the point where 63.2% percent of the population will have failed, regardless of the shape parameter (β). The following shape parameter characteristics are noted:

β = 1.0 : Exponential distribution, constant failure rate
β = 3.5 : Normal distribution (approximation)
β < 1.0 : Decreasing failure (hazard) rate
β > 1.0 : Increasing failure (hazard) rate

The calculation inputs below show units of "hours," but any life metric (cycles, years, etc.) can be used as long as there is consistency among the three inputs.

Calculation Inputs:

1. Shape parameter (β):
2. Chacteristic life (η, hours):
3. Time period of interest (t, hours):
4. Decimal places:
5. Options:

Calculate mean life
f(t) chart
R(t) chart
h(t) chart


Python Shell. Can be used as a calculator.

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  1. MIL-HDBK-338, Electronic Reliability Design Handbook.
  2. Bazovsky, Igor, Reliability Theory and Practice.
  3. O'Connor, Patrick, D. T., Practical Reliability Engineering.
  5. Weibull Distribution, NIST Engineering Statistics Handbook .
  6. Barringer, Paul, Typical beta (β) values: