Weibull Distribution
The Weibull distribution requires two parameters, characteristic life and the shape factor, which is also known as the slope.
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Special cases of the Weibull distribution use as few as one parameter and as many as three parameters. The special case that uses only one parameter is known as the WeiBayes distribution as indicated in WeiBayes Distribution. The special case that uses a third parameter, uses a location parameter to shift the origin of the age scale. For more information, see Location Parameter
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Among all the distributions available for reliability calculations, the Weibull demonstration is the only one unique to the field. In his original paper, A Distribution of Wide Applicability, Professor Weibull, who was studying metallurgical failures, argued that normal distributions require that initial metallurgical strengths be normally distributed, and that what was needed was a function that could embrace a great variety of distributions (including the normal).
The two-parameter Weibull distribution provides reasonably accurate failure analysis and failure forecasts with extremely small samples. It has the special capability to diagnose failure types, such as infant mortality (particularly for electronics), age-independent failures (accidents and natural occurrences), and wear-out type mechanisms (bearings, filters, and so on.). The two-parameter Weibull distribution is recommended if the failure rate decreases (burn-in period) or increases (wear-out period) over time, or if the failure rate remains constant (random failure period).
The shape factor value describes the shape of the hazard curve. Examples follow.
• When the shape factor is < 1, the curve takes on a gamma distribution (approximately).
• When the shape factor is = 1, the curve takes on an exponential distribution. This is the constant failure rate curve.
• When the shape factor is = 2, the curve takes on a lognormal distribution (approximately).
• When the shape factor is = 3.5, the curve takes on a normal distribution (approximately).
Calculations
The probability density function, f(t), and the survival function, that is reliability, 
, with respect to time t, follows for computing Weibull distributions.
, with respect to time t, follows for computing Weibull distributions.
Where:
, 
, and 
In this case, β and η are shape and scale (characteristic life) parameters of the distribution.