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Weibull Distribution Mean Formula:
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The Weibull distribution is a continuous probability distribution commonly used in reliability engineering, failure analysis, and survival analysis. The mean of the Weibull distribution represents the expected value or average of the distribution.
The calculator uses the Weibull distribution mean formula:
Where:
Explanation: The scale parameter η represents the characteristic life, while the shape parameter β determines whether the distribution is exponential (β=1), Rayleigh (β=2), or approximates normal distribution (β=3-4).
Details: The Weibull distribution is widely used in reliability engineering to model time-to-failure data, in weather forecasting for wind speed distributions, and in medical research for survival analysis.
Tips: Enter the scale parameter η and shape parameter β. Both values must be positive numbers. The calculator will compute the mean using the gamma function.
Q1: What does the shape parameter β indicate?
A: β < 1 indicates decreasing failure rate, β = 1 indicates constant failure rate (exponential), β > 1 indicates increasing failure rate.
Q2: What is the physical meaning of the scale parameter η?
A: η represents the characteristic life where approximately 63.2% of units have failed.
Q3: When is the Weibull distribution used?
A: Commonly used in reliability analysis, failure prediction, wind energy assessment, and survival analysis in medical research.
Q4: What are typical values for η and β?
A: η depends on the application (could be hours, cycles, etc.), β typically ranges from 0.5 to 5 depending on the failure mechanism.
Q5: How accurate is the gamma function calculation?
A: The calculator uses Lanczos approximation which provides high accuracy for most practical applications.