Weibull Reliability R(t) = exp(−(t/η)^β), Hazard Rate h(t) & B10 Life Calculator

Model infant mortality (β<1), random failure (β=1), and wear-out (β>1) Weibull reliability.

Weibull Survival Reliability R(2500)
80.442 %
Cumulative Failure Probability F(t) 19.558 %
B10 Life (Time to 10% Failure) 1,797.764 hours
Hazard Rate per 1,000 Hours 0.192 failures/10³h
Inputs
⚡ Scenarios:
β
Weibull shape slope β (<1 infant mortality, 1.0 constant hazard, >1 wear-out).
hours η
Characteristic life scale parameter η (time at which 63.2% have failed).
hours t
Target operating mission time t.
Results & Graph
Weibull Survival Reliability R(2500)
80.442 %
Cumulative Failure Probability F(t)
19.558 %
B10 Life (Time to 10% Failure)
1,797.764 hours
Hazard Rate per 1,000 Hours
0.192 failures/10³h

Weibull Survival Reliability R(2500)

Graph 1 • Weibull Reliability & B10 Life • Domain Architecture & Live Telemetry Maps Shape Parameter Beta, Scale Characteristic Life Eta, Mission Operating Time T through Weibull Reliability & B10 Life Formula.
Weibull Survival Reliability R: 80.442 % Weibull Reliability & B10 Life • Invariants & Metrics Weibull Survival Reliability 80.442 % Cumulative Failure Probabili 19.558 % B10 Life (Time to 10% Failur 1,797.764 hours Hazard Rate per 1,000 Hours 0.192 failures/10³h
Graph 2 • Weibull Survival Reliability R(2500) Across Shape Parameter Beta Sweep (40% → 160%) Evaluates R(t) = exp( −(t / η)^β ); h(t) = (β / η) × (t / η)^(β − 1); B10_Life across 7 operating points (current input highlighted).
0 % 50.41 % 100.8 % 58.08 0.880β 67.00 1.32β 74.44 1.76β 80.44 2.20β 85.18 2.64β 88.85 3.08β 91.65 3.52β
Graph 3 • Joint Sensitivity: Shape Parameter Beta × Scale Characteristic Life Eta Compares Weibull Survival Reliability R(2500) when Scale Characteristic Life Eta is shifted +25% (solid) vs. −25% (dashed).
0 % 51.14 % 102.3 % --- Scale Characteristic Life Eta @ 75% 78.36 -30% 83.51 -15% 87.53 +0% 90.62 +15% 92.98 +30%
Summary: At t = 2500 hrs (β=2.2, η=5000 hrs), survival reliability R(t) is 80.44%, cumulative failure F(t) is 19.56%, and B10 life (10% failure) is 1797.8 hours.
Summary: At t = 2500 hrs (β=2.2, η=5000 hrs), survival reliability R(t) is 80.44%, cumulative failure F(t) is 19.56%, and B10 life (10% failure) is 1797.8 hours.
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Formula & Mathematical Methodology: Weibull Reliability & B10 Life Formula

R(t) = exp( −(t / η)^β )
h(t) = (β / η) × (t / η)β − 1
B10Life = η × (−ln(0.90))1/β
R(t) = exp( −(t / η)^β ); h(t) = (β / η) × (t / η)^(β − 1); B10_Life = η × (−ln(0.90))^(1/β)
  • shapeParameterBeta: Shape Parameter Beta (β)
  • scaleCharacteristicLifeEta: Scale Characteristic Life Eta (hours η)
  • missionOperatingTimeT: Mission Operating Time T (hours t)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Weibull Reliability & B10 Life

Model infant mortality (β<1), random failure (β=1), and wear-out (β>1) Weibull reliability.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: R(t) = exp( −(t / η)^β ); h(t) = (β / η) × (t / η)^(β − 1); B10_Life = η × (−ln(0.90))^(1/β).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Weibull Reliability & B10 Life via R(t) = exp( −(t / η)^β ); h(t) = (β / η) × (t / η)^(β − 1); B10_Life = η × (−ln(0.90))^(1/β).

Frequently Asked Questions

How does the Weibull Reliability R(t) = exp(−(t/η)^β), Hazard Rate h(t) & B10 Life Calculator compute the solution?

It evaluates R(t) = exp( −(t / η)^β ); h(t) = (β / η) × (t / η)^(β − 1); B10_Life = η × (−ln(0.90))^(1/β) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Probability Distributions & Survival Analysis

Weibull Reliability R(t) = exp(−(t/η)^β), Hazard Rate h(t) & B10 Life Calculator: inputs, method, and how to read the result

Model infant mortality (β<1), random failure (β=1), and wear-out (β>1) Weibull reliability.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Shape Parameter Beta: Weibull shape slope β (<1 infant mortality, 1.0 constant hazard, >1 wear-out).
  • Scale Characteristic Life Eta: Characteristic life scale parameter η (time at which 63.2% have failed).
  • Mission Operating Time T: Target operating mission time t.

What the calculation includes

With the default values shown, Weibull Survival Reliability R(2500) is 80.442 %. The calculation follows Weibull Reliability & B10 Life Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Shape Parameter Beta from 2.2 β to 2.42 β changes Weibull Survival Reliability R(2500) from 80.442 % to 82.956 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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