0.551819
0.632121
0.9027
0.3757
0.551819
0.632121
0.9027
0.3757
The Weibull Distribution Calculator computes the probability density function (PDF), cumulative distribution function (CDF), and key moments of the Weibull distribution. Named after Swedish mathematician Waloddi Weibull, this distribution is one of the most widely used models in reliability engineering, survival analysis, and failure-time modeling. Its remarkable flexibility allows it to model increasing, decreasing, or constant failure rates depending on the shape parameter.
The Weibull distribution is defined by two parameters: the shape parameter k (also called the Weibull modulus) and the scale parameter λ. The shape parameter determines the failure rate behavior: when k < 1, the failure rate decreases over time (infant mortality), when k = 1, the failure rate is constant (random failures, equivalent to the exponential distribution), and when k > 1, the failure rate increases over time (wear-out failures). This three-regime behavior makes the Weibull distribution the backbone of the bathtub curve in reliability theory.
In practice, the Weibull distribution is used across numerous industries. Manufacturing engineers use it to predict product lifetimes and set warranty periods. Wind energy analysts model wind speed distributions with Weibull parameters to estimate power generation potential. Materials scientists characterize the strength of brittle materials (ceramics, composites) using Weibull statistics. Biostatisticians employ it in survival analysis to model time-to-event data such as patient survival times or disease recurrence intervals.
The scale parameter λ represents the characteristic life — the time at which approximately 63.2% of units will have failed (since CDF at x = λ always equals 1 − e⁻¹ ≈ 0.632 regardless of k). Engineers frequently report the B10 life (time at which 10% of units fail) and B50 life (median), both easily derivable from the Weibull CDF. The hazard function h(x) = (k/λ)(x/λ)^(k−1) is monotonically increasing for k > 1, constant for k = 1, and decreasing for k < 1, which directly connects to the physical failure mechanism.
This calculator computes the mean and variance using the gamma function evaluated at arguments depending on k, along with the PDF and CDF at any specified point. These results enable reliability engineers to assess product lifetime characteristics, estimate warranty costs, and design accelerated life testing programs.
The Weibull distribution PDF and CDF are:
$$f(x; k, \lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}, \quad x \geq 0$$
$$F(x; k, \lambda) = 1 - e^{-(x/\lambda)^k}$$
The mean and variance involve the gamma function:
$$\mu = \lambda \cdot \Gamma\left(1 + \frac{1}{k}\right), \quad \sigma^2 = \lambda^2 \left[\Gamma\left(1 + \frac{2}{k}\right) - \Gamma^2\left(1 + \frac{1}{k}\right)\right]$$
The CDF formula is particularly useful because it yields the reliability function R(x) = 1 − F(x) = e^(−(x/λ)^k), which gives the probability of survival beyond time x.
The PDF shows the relative likelihood at each point — the peak indicates the most probable failure time when k > 1. The CDF gives the cumulative probability of failure by time x; equivalently, 1 − CDF is the survival (reliability) probability. A CDF value of 0.1 at time x means 10% of units have failed by that time. The mean is the average expected lifetime, and the variance indicates the spread of lifetimes. Higher shape parameters yield more concentrated distributions around the mean, while lower values indicate greater variability in failure times.
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Results
With shape=2.5 and scale=5000 hours, about 22.6% of bearings fail by 3000 hours. The mean lifetime is approximately 4431 hours with increasing failure rate (k > 1).
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Results
Weibull(2, 8) models wind speeds with mean ~7.1 m/s. At 6 m/s, the CDF is 0.43, meaning 43% of observations are below this speed.
The shape parameter directly determines the failure rate trend: k < 1 means the failure rate decreases over time (early-life failures or infant mortality), k = 1 gives a constant failure rate (random failures, equivalent to exponential distribution), and k > 1 means the failure rate increases over time (wear-out, aging, fatigue). Most engineered products exhibit k between 1.5 and 4, indicating wear-out failure modes. Values of k > 3.5 indicate very concentrated failure times, typical of fatigue-limited components.
The characteristic life is the scale parameter λ, defined as the time at which 63.2% of units have failed (CDF = 0.632). This holds regardless of the shape parameter. It serves as a standardized reference point for comparing different Weibull distributions. In reliability specifications, the B10 life (time at which 10% fail) is often more practically useful and can be calculated as λ × (−ln(0.9))^(1/k).
Wind speeds at a given location are commonly modeled by a Weibull distribution with shape parameter k typically between 1.5 and 3, and scale parameter proportional to the mean wind speed. This model allows engineers to estimate the probability distribution of power output, calculate capacity factors, and determine the economic viability of wind farm sites. The two-parameter Weibull fits observed wind speed data well in most locations and is the standard model in wind resource assessment.
Common methods include: Maximum Likelihood Estimation (MLE), which is most efficient for large samples; Weibull probability plotting, which linearizes the CDF and fits a straight line to get k (slope) and λ (intercept); and the Method of Moments, matching sample mean and variance to theoretical formulas. For censored data (common in reliability testing where not all units have failed), MLE with appropriate likelihood modifications is the standard approach.
The exponential distribution is a special case of the Weibull with k = 1. When k = 1, the PDF simplifies to (1/λ)e^(−x/λ), which is exactly the exponential distribution with rate 1/λ. The exponential's memoryless property (constant hazard rate) is unique to k = 1. Any deviation of k from 1 introduces time-dependent failure rates, which is why the Weibull generalizes the exponential for non-constant hazard scenarios.
Yes. When k < 1, the Weibull hazard rate h(x) = (k/λ)(x/λ)^(k−1) is a decreasing function of time. This models infant mortality or burn-in failure patterns, where weak units fail early and survivors become increasingly reliable. This behavior is observed in electronic components during initial operation and in some biological systems. The bathtub curve in reliability combines a Weibull (k < 1) for early failures, exponential for random mid-life failures, and Weibull (k > 1) for wear-out.
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