What is the characteristic life eta?

The scale parameter eta, also called the characteristic life, is the time at which 63.2% of the population has failed. This is because F(eta) = 1 - e^(-1) = 0.632 regardless of the value of beta. It serves as a natural scale for the distribution and is used as a reference point in reliability analysis.

Weibull Distribution Calculator

Calculate Weibull distribution probabilities, reliability R(t), hazard rate h(t), and B-life percentiles. Enter shape β and scale η to get PDF, CDF, mean, variance, MTTF, and step-by-step solutions with interactive graphs showing the bathtub curve behavior.

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About Weibull Distribution Calculator

The Weibull Distribution Calculator computes probabilities, reliability, hazard rates, and key statistics for the Weibull distribution $X \sim \text{Weibull}(\beta, \eta)$. Enter the shape parameter $\beta$ and scale parameter $\eta$, and get the failure probability $F(x)$, reliability $R(x)$, hazard function $h(x)$, B-life percentiles, and a step-by-step solution with interactive PDF, CDF, and hazard function graphs. This tool is essential for reliability engineering, survival analysis, and lifetime data modeling.

What Is the Weibull Distribution?

The Weibull distribution is a continuous probability distribution named after Swedish mathematician Waloddi Weibull. It is the most widely used distribution in reliability engineering and life data analysis because its shape parameter $\beta$ allows it to model three distinct failure behaviors: decreasing failure rate (infant mortality), constant failure rate (random failures), and increasing failure rate (wear-out). The probability density function is:

$$f(x) = \frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1} e^{-(x/\eta)^\beta}, \quad x \geq 0$$

The Shape Parameter β and the Bathtub Curve

The shape parameter $\beta$ (beta) determines the failure rate behavior and directly relates to the bathtub curve used in reliability engineering:

📉

β < 1: Infant Mortality

Decreasing failure rate. Early-life defects are weeded out over time. Common in electronics burn-in.

➡

β = 1: Random Failures

Constant failure rate. Reduces to exponential distribution. Memoryless property applies.

📈

β > 1: Wear-Out

Increasing failure rate. Aging and degradation dominate. Common in mechanical components.

🔔

β ≈ 3.6: Normal-Like

Approximates a symmetric normal distribution. Useful for fatigue life modeling.

Key Formulas

Property	Formula	Description
PDF	$\frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1} e^{-(x/\eta)^\beta}$	Probability density at x
CDF	$F(x) = 1 - e^{-(x/\eta)^\beta}$	Failure probability by time x
Reliability	$R(x) = e^{-(x/\eta)^\beta}$	Survival probability at time x
Hazard	$h(x) = \frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1}$	Instantaneous failure rate
Mean	$\eta \cdot \Gamma(1 + 1/\beta)$	Mean time to failure (MTTF)
Variance	$\eta^2[\Gamma(1+2/\beta) - \Gamma^2(1+1/\beta)]$	Spread of lifetime
Median	$\eta(\ln 2)^{1/\beta}$	50th percentile life
Mode	$\eta\left(\frac{\beta-1}{\beta}\right)^{1/\beta}$ for β > 1	Most probable lifetime
B-Life	$\eta(-\ln(1-p))^{1/\beta}$	Time for p fraction to fail
Char. Life	$\eta$ → F(η) = 63.2%	Scale parameter interpretation

Real-World Applications

Industry	Application	Typical β
Aerospace	Turbine blade fatigue life	2 – 4
Automotive	Bearing wear-out analysis	1.5 – 3
Electronics	Semiconductor infant mortality	0.3 – 0.8
Power Systems	Wind speed distribution	1.5 – 3
Medical Devices	Implant survival time	1.5 – 5
Manufacturing	Warranty planning and B10 life	1.5 – 4
Civil Engineering	Concrete and material strength	5 – 20

Weibull vs. Other Distributions

Feature	Weibull	Exponential	Lognormal
Parameters	β (shape), η (scale)	λ (rate)	μ, σ
Failure Rate	Flexible (↓, →, ↑)	Constant only	Rises then falls
Special Case	β=1 → Exponential	Weibull β=1	—
Best For	Mechanical wear-out	Random events	Repair times
B-Life Analysis	Native support	Limited	Possible

How to Use the Weibull Distribution Calculator

Enter the shape parameter β: This controls the failure rate behavior. Use β < 1 for infant mortality, β = 1 for constant failure rate (exponential), or β > 1 for wear-out failures. Common values range from 0.5 to 5. The real-time insight badge shows you what your β value means.
Enter the scale parameter η: This is the characteristic life — the time at which 63.2% of units have failed. It sets the time scale for the distribution. For example, if a bearing has η = 5000 hours, then 63.2% of bearings fail by 5000 hours.
Select the probability type: Choose P(X ≤ x) for failure probability, R(x) = P(X > x) for reliability (survival probability), or P(a ≤ X ≤ b) for range probability.
Enter the time value: Enter the time, cycles, or usage value. For range mode, enter both lower and upper bounds.
Review the results: Examine the probability, animated probability bar, interactive PDF/CDF/hazard function graphs, reliability milestones (MTTF, B1, B10 life), distribution properties, and the complete step-by-step solution with MathJax formulas.

FAQ

What is the Weibull distribution?

The Weibull distribution is a continuous probability distribution used extensively in reliability engineering and failure analysis. It is defined by two parameters: shape β and scale η. Its flexibility allows it to model decreasing, constant, or increasing failure rates depending on the value of β, making it the standard tool for lifetime data analysis across aerospace, automotive, electronics, and many other industries.

What does the shape parameter β mean?

The shape parameter β (beta) controls the failure rate behavior. When β is less than 1, the failure rate decreases over time, which models infant mortality or early-life failures. When β equals 1, the failure rate is constant and the Weibull reduces to an exponential distribution. When β is greater than 1, the failure rate increases over time, modeling wear-out and aging. A β around 3.6 approximates a normal distribution, which is useful for symmetric fatigue life modeling.

What is B10 life in Weibull analysis?

B10 life (also written as B₁₀ or L₁₀) is the time at which 10% of the population is expected to have failed, meaning 90% survive. It is a critical metric in reliability engineering used for warranty planning, maintenance scheduling, and specification compliance. Similarly, B1 life means 1% failure. These B-life values are calculated from the inverse of the Weibull CDF: t = η × (−ln(1−p))^(1/β).

What is the relationship between Weibull and exponential distributions?

The exponential distribution is a special case of the Weibull distribution when the shape parameter β equals 1. In this case, the failure rate is constant over time, which corresponds to the memoryless property. The Weibull generalizes the exponential by allowing the failure rate to change over time — decreasing for β less than 1 and increasing for β greater than 1.

What is the characteristic life η?

The scale parameter η (eta), also called the characteristic life, is the time at which exactly 63.2% of the population has failed. This holds regardless of the value of β because F(η) = 1 − e^(−1) ≈ 0.632. In reliability engineering, η serves as a natural reference point and time scale for the distribution.

Reference this content, page, or tool as:

"Weibull Distribution Calculator" at https://MiniWebtool.com/weibull-distribution-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-14

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About Weibull Distribution Calculator

What Is the Weibull Distribution?

The Shape Parameter β and the Bathtub Curve

Key Formulas

Real-World Applications

Weibull vs. Other Distributions

How to Use the Weibull Distribution Calculator

FAQ

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Property	Formula	Description
PDF	\(\frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1} e^{-(x/\eta)^\beta}\)	Probability density at x
CDF	\(F(x) = 1 - e^{-(x/\eta)^\beta}\)	Failure probability by time x
Reliability	\(R(x) = e^{-(x/\eta)^\beta}\)	Survival probability at time x
Hazard	\(h(x) = \frac{\beta}{\eta}\left(\frac{x}{\eta}\right)^{\beta-1}\)	Instantaneous failure rate
Mean	\(\eta \cdot \Gamma(1 + 1/\beta)\)	Mean time to failure (MTTF)
Variance	\(\eta^2[\Gamma(1+2/\beta) - \Gamma^2(1+1/\beta)]\)	Spread of lifetime
Median	\(\eta(\ln 2)^{1/\beta}\)	50th percentile life
Mode	\(\eta\left(\frac{\beta-1}{\beta}\right)^{1/\beta}\) for β > 1	Most probable lifetime
B-Life	\(\eta(-\ln(1-p))^{1/\beta}\)	Time for p fraction to fail
Char. Life	\(\eta\) → F(η) = 63.2%	Scale parameter interpretation