Calculate Weibull distribution PDF, CDF, survival, hazard rate, MTTF, percentiles, and B-life. Supports reliability analysis with shape and scale parameters plus mission time planning.
The Weibull distribution calculator computes probabilities, reliability metrics, and failure analysis for the Weibull distribution — the most widely used model in reliability engineering. By adjusting two parameters (shape k and scale λ), it can model infant mortality failures, random failures, and wear-out failures.
The shape parameter (k) determines the failure behavior: k < 1 models decreasing hazard rate (infant mortality), k = 1 reduces to the exponential distribution (constant hazard), and k > 1 models increasing hazard (wear-out). The scale parameter (λ) is the "characteristic life" — the time at which 63.2% of units have failed.
Enter parameters and values to compute PDF, CDF, survival probability, hazard rate, MTTF, percentiles, B-life, and reliability at a specific mission time. The visual PDF curve and reliability table provide intuitive understanding of the failure distribution. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
The Weibull distribution is the most important distribution in reliability engineering and failure analysis. With just two parameters, it flexibly models infant mortality, random failures, and wear-out — making it essential for product design, warranty analysis, maintenance planning, and safety engineering.
This calculator provides all the key metrics engineers need: MTTF, B-life, reliability at a given time, hazard rate trends, and percentile tables.
PDF: f(x) = (k/λ)(x/λ)^(k−1) e^(−(x/λ)^k). CDF: F(x) = 1 − e^(−(x/λ)^k). Hazard: h(x) = (k/λ)(x/λ)^(k−1). Mean: λΓ(1+1/k). Median: λ(ln2)^(1/k).
Result: F(5) = 22.12%, R(8) = 52.73%, MTTF = 8.862
With k=2 (increasing hazard) and λ=10, 22.12% fail by time 5. Reliability at time 8 is 52.73%. Mean time to failure is 8.862. The hazard rate increases linearly (k=2 gives linear hazard), modeling typical wear-out failure.
In practice, engineers collect failure time data, create a Weibull probability plot, estimate the shape and scale parameters (via maximum likelihood or linear regression), and then use the fitted distribution for predictions. Good parameters from Weibull analysis can predict warranty returns, optimize maintenance schedules, and set reliability targets for design improvements.
Product failure rates typically follow a "bathtub" pattern: high initially (infant mortality from manufacturing defects), constant during useful life (random failures), and increasing again (wear-out). Each phase is modeled by a different Weibull shape parameter. Reliability programs use burn-in testing to eliminate infant mortality, preventive maintenance to catch wear-out, and redundancy to mitigate random failures.
When k = 1, Weibull reduces to the exponential distribution with rate 1/λ. When k = 2, it's the Rayleigh distribution (used for wind speed and signal fading). When k ≈ 3.44, it closely matches the normal distribution. This flexibility is why the Weibull is called the "universal" distribution in reliability.
The shape parameter k determines how the failure rate changes over time. k < 1: failures decrease over time (infant mortality — defective units fail early). k = 1: constant failure rate (random failures). k > 1: failures increase over time (wear-out). k ≈ 3.5 gives a bell-shaped PDF similar to normal.
The scale parameter λ is the characteristic life — the time at which 63.2% of units have failed (CDF = 63.2%). This property holds regardless of the shape parameter. It's analogous to the mean of an exponential distribution when k = 1.
Weibull analysis uses failure data to estimate k and λ, then predicts warranty costs, maintenance schedules, and reliability targets. Engineers use Weibull plots (log-log paper) to estimate parameters and check model fit. It models everything from ball bearings to electronics.
B-life is the time at which a specified percentage of units have failed. B10 means 10% failed, B50 is the median life. In bearing engineering, B10 life is a standard specification. The formula is t = λ(−ln(1−p))^(1/k).
The bathtub curve combines three Weibull phases: infant mortality (k < 1) in early life, random failures (k ≈ 1) during useful life, and wear-out (k > 1) in late life. A single Weibull models one phase; the full bathtub requires a mixture of Weibulls.
Yes! Wind speed at any location approximately follows a Weibull distribution with k ≈ 2 (Rayleigh), where the scale parameter relates to average wind speed. Wind energy production calculations use Weibull probability to estimate power output over a year.