The Weibull distribution has widespread use in life data analysis. It is a flexible distribution able to model decreasing or increasing failure rates, and approximate many other distributions including exponential and normal.
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The shape parameter, β, and the scale parameter, η, fully defines the Weibull distribution.
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F(t) shows the probability of failure before a given point in time, t.
The scale parameter, η, is the point in time when 63.2% have failed. Use the slider for the scale and notice as the scale increases it take longer for the same number of units to fail.
The shape of the curve is represented by shape parameter, β. When β < 1 the curve is initially steeper then decreases the rate of additional failures. When β = 1 the curve maintains a steady rate of increase in percentile of failure with time. When β >1 the curve initially has a low probability of failure then ramps up to an inflection point at the 63.2 percentile then has a decreasing rate of adding failures.
Use the slider for the shape parameter, and notice higher values steepen the curve. Lower values make the CDF curve increasingly flat.
The value, F(t), is defined for t > 0 and returns values between 0 and 1. The x-axis is time in this example. It could be length, weight, temperature, or any continuous variable.
The Weibull CDF is commonly used in reliability work to describe the time to failure pattern for a wide range of failure mechanisms. The quick Interpretation of \[Beta] (decreasing, constant, or increasing failure rate) make the Weibull distribution a common first look at the data.
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