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Weibull Distribution Calculator

Ready to calculate
Weibull PDF & CDF.
MTTF & Reliability.
Hazard Rate Analysis.
100% Free.
No Data Stored.

How it Works

01Enter Shape (k) & Scale (λ)

Shape k controls failure rate behavior; scale λ is the characteristic life.

02Enter Time Point x

The time at which to compute failure probability and reliability.

03Get CDF & Reliability

F(x) = failure probability; R(x) = survival probability at time x.

04MTTF & Hazard Rate

Mean time to failure and instantaneous hazard rate at time x.

Introduction

The Weibull distribution is one of the most versatile and widely used probability distributions in reliability engineering, survival analysis, and failure time modeling. The Weibull distribution calculator computes the probability density function (PDF), cumulative distribution function (CDF), reliability function, hazard rate, mean, variance, and quantiles for any combination of shape (k) and scale (λ) parameters.

Named after Swedish mathematician Waloddi Weibull, who popularized it in his 1951 paper on statistical distribution functions of wide applicability, the Weibull distribution can model a remarkable range of failure behaviors. By adjusting the shape parameter k, it can represent decreasing failure rates (infant mortality, k<1), constant failure rates (random failures, k=1, giving the exponential distribution), or increasing failure rates (wear-out failures, k>1).

The three failure rate regimes correspond to the three periods of the famous "bathtub curve" in reliability engineering: infant mortality (early life failures due to manufacturing defects), useful life (constant random failures), and wear-out (aging and fatigue failures). The Weibull distribution models any of these with appropriate parameter choices.

Applications span: engineering reliability analysis, survival analysis in medicine and biology, wind speed and precipitation modeling in meteorology, financial risk modeling, material strength testing, quality control and product lifetime analysis, and warranty claims forecasting.

This calculator enables complete Weibull analysis: computing probabilities, survival probabilities, median lifetime, mean time to failure (MTTF), and percentile values — all essential for making reliability predictions and maintenance decisions.

The formula

Parameters:
  • k = shape parameter (k > 0)

  • λ = scale parameter (λ > 0, the characteristic life)

    • PDF:
    f(x) = (k/λ) × (x/λ)^(k−1) × e^−(x/λ)^k

    CDF (Failure Probability):
    F(x) = 1 − e^−(x/λ)^k

    Reliability (Survival):
    R(x) = e^−(x/λ)^k

    Hazard Rate:
    h(x) = (k/λ) × (x/λ)^(k−1)

    Mean (MTTF):
    E = λ × Γ(1 + 1/k)

    Variance:
    Var = λ² × [Γ(1+2/k) − (Γ(1+1/k))²]

    Real-World Example

    Calculation In Practice

    Example: Ball Bearing Lifetime
    Shape k = 2.5 (wear-out failures)
    Scale λ = 1000 hours

    Probability of failure by 800 hours:
    F(800) = 1 − e^−(800/1000)^2.5
    = 1 − e^−(0.8)^2.5
    = 1 − e^−0.572
    = 1 − 0.564
    = 0.436 (43.6%)

    Reliability at 800 hours: R(800) = 56.4%

    MTTF = 1000 × Γ(1+1/2.5) = 1000 × Γ(1.4) ≈ 887 hours

    Typical Use Cases

    1

    Product Reliability Analysis

    Model product failure times to estimate mean time to failure (MTTF) and warranty durations.
    2

    Survival Analysis

    Model time-to-event data in clinical trials for cancer survival, equipment failure, and customer churn.
    3

    Wind Energy Modeling

    Weibull distribution models wind speed distributions for turbine energy output estimation.
    4

    Maintenance Planning

    Compute failure probabilities to schedule preventive maintenance at optimal intervals.
    5

    Material Strength Testing

    Model fracture strength and fatigue life of materials under varying stress conditions.

    Technical Reference

    Shape Parameter k Interpretation:
  • k < 1: decreasing failure rate (infant mortality)

  • k = 1: constant failure rate (exponential distribution, random failures)

  • 1 < k < 4: increasing failure rate (wear-out begins)

  • k ≈ 3.4: approximates normal distribution

  • k ≥ 4: rapid wear-out phase

    • Median Life:
    T₅₀ = λ × (ln 2)^(1/k)

    B10 Life (10th percentile):
    T₁₀ = λ × (−ln 0.9)^(1/k)

    Parameter Estimation:

  • MLE: maximize log-likelihood function

  • Regression: plot ln(−ln(1−F)) vs ln(t) → slope = k, intercept = −k×ln(λ)

    • Bathtub Curve:

  • Burn-in: k < 1 (DFR)

  • Useful life: k = 1 (CFR)

  • Wear-out: k > 1 (IFR)

    • Key Takeaways

    The Weibull distribution is the workhorse of reliability engineering precisely because its shape parameter k gives it such flexibility: it can represent decreasing, constant, or increasing hazard rates with a single two-parameter family. This makes it applicable to a vast range of physical phenomena, from electronic component failure to human disease progression to structural fatigue.

    Key values to remember: k=1 gives the exponential distribution (constant hazard, random failures). k=2 gives the Rayleigh distribution (common in wave height modeling). k≈3.4 approximates the normal distribution for symmetrical wear-out processes.

    For practical reliability analysis, Weibull parameters are typically estimated from failure time data using maximum likelihood estimation (MLE) or Weibull probability paper regression. Once parameters are estimated, this calculator provides all the reliability metrics needed for engineering decisions.

    Frequently Asked Questions

    What is the Weibull distribution used for?
    The Weibull distribution models time-to-failure data in reliability engineering, survival analysis, wind speed, and any continuous process where the failure rate changes over time.
    What does the shape parameter k mean?
    k determines the failure rate behavior: k<1 means failures decrease over time (infant mortality); k=1 means constant failure rate (exponential); k>1 means failures increase over time (wear-out).
    What does the scale parameter lambda mean?
    λ (lambda) is the characteristic life — the time at which approximately 63.2% of items have failed (since F(λ) = 1−e⁻¹ ≈ 0.632), regardless of the shape parameter.
    How is the Weibull related to the exponential distribution?
    When k=1, the Weibull distribution reduces to the exponential distribution with rate 1/λ. The exponential distribution has constant hazard rate — a special case of Weibull.
    What is MTTF?
    Mean Time to Failure: E = λ × Γ(1+1/k). It is the expected lifetime of a component before first failure, computed using the gamma function.
    What is the reliability function?
    R(t) = 1 − F(t) = e^−(t/λ)^k. It gives the probability that a component survives beyond time t — the survival function for the Weibull distribution.
    What is the hazard rate in the Weibull model?
    h(t) = (k/λ)(t/λ)^(k−1). It represents the instantaneous failure rate at time t given survival to time t. For k>1, this increases over time (wear-out).
    What is the bathtub curve?
    The bathtub curve shows how product failure rates change over lifetime: high during infant mortality (k<1), constant during useful life (k=1), and increasing during wear-out (k>1). The Weibull models each segment.
    How do I estimate Weibull parameters from data?
    Use maximum likelihood estimation (MLE) with statistical software, or Weibull probability paper regression: plot ln(−ln(1−F̂)) vs ln(t). The slope gives k and the intercept gives −k×ln(λ).
    What is B10 life?
    B10 life (or L10) is the time at which 10% of items have failed: T₁₀ = λ×(−ln(0.9))^(1/k). It is commonly specified for bearings and other mechanical components.

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