Weibull Distribution
Compute Weibull distribution PDF, CDF, mean, median, mode, variance, and skewness online. Free probability calculator with interactive PDF and CDF charts.
About This Calculator
The Weibull Distribution Calculator computes the probability density function (PDF), cumulative distribution function (CDF), and key statistical measures including mean, median, mode, variance, standard deviation, and skewness for the Weibull distribution given the shape parameter k and scale parameter λ. This free online tool is ideal for students, statisticians, reliability engineers, and researchers working in survival analysis and failure modeling.
How It Works
Enter the shape parameter k (k > 0) and scale parameter λ (λ > 0) of the Weibull distribution, along with the value x (x ≥ 0) at which to evaluate the PDF and CDF. The calculator uses the standard Weibull formulas: PDF f(x) = (k/λ)(x/λ)^(k-1)e^(-(x/λ)^k) and CDF F(x) = 1 - e^(-(x/λ)^k). All statistical moments are derived using the gamma function Γ, where the r-th moment is λ^r × Γ(1 + r/k).
Key computed measures include Mean = λ × Γ(1 + 1/k), Median = λ × (ln 2)^(1/k), Mode = λ × ((k-1)/k)^(1/k) for k > 1, Variance = λ² × [Γ(1 + 2/k) - Γ(1 + 1/k)²], and Skewness derived from the third central moment. The interactive chart displays the PDF or CDF curve across a range of x values, updating dynamically with parameter changes.
Regional Notes
India: The Weibull distribution is widely used in Indian engineering and manufacturing sectors for reliability analysis and quality control under BIS standards. It is also applied in actuarial science by IRDAI-regulated insurance companies for mortality and claims modeling.
United States: The Weibull distribution is extensively used in US defense and aerospace for MIL-STD reliability testing, in FDA-regulated medical device failure analysis, and in NIST statistical process control applications. The shape parameter interpretation guides maintenance scheduling in manufacturing.
United Kingdom: UK researchers apply the Weibull distribution in wind energy analysis through the Renewable Energy Directive, in HSE workplace risk assessment for equipment failure rates, and by the ONS for demographic survival modeling and life expectancy calculations.
Frequently Asked Questions
What is the Weibull distribution?
The Weibull distribution is a continuous probability distribution widely used in reliability engineering, survival analysis, and failure analysis. It is defined by a shape parameter k and a scale parameter λ, and can model various distribution shapes from exponential to nearly symmetric bell-shaped depending on the value of k.
What does the shape parameter k mean in the Weibull distribution?
The shape parameter k determines the behavior of the Weibull distribution. When k < 1, the failure rate decreases over time (infant mortality). When k = 1, the distribution reduces to the exponential distribution with a constant failure rate. When k > 1, the failure rate increases over time (wear-out failures). As k increases beyond 3, the distribution becomes approximately symmetric.
How do you calculate the mean of the Weibull distribution?
The mean of the Weibull distribution is calculated as λ × Γ(1 + 1/k), where Γ is the gamma function. For example, with shape k = 2 and scale λ = 1, the mean is approximately 0.8862. The median is λ × (ln 2)^(1/k), and the mode is λ × ((k-1)/k)^(1/k) when k > 1.
What is the Weibull distribution used for?
The Weibull distribution is used in reliability engineering to model time-to-failure data, in survival analysis to study patient survival times, in weather forecasting to model wind speed distributions, in hydrology for flood frequency analysis, and in quality control for process capability analysis. Its flexibility in modeling increasing, decreasing, and constant failure rates makes it invaluable across these fields.
What is the difference between the Weibull and normal distributions?
The Weibull distribution differs from the normal distribution in several key ways. Weibull only models non-negative values while normal covers all real numbers. Weibull can model increasing, decreasing, or constant failure rates, whereas the normal distribution is always symmetric. With k ≈ 3.4, the Weibull distribution closely approximates a normal distribution in shape.
What is the PDF formula for the Weibull distribution?
The probability density function (PDF) of the Weibull distribution is f(x) = (k/λ) × (x/λ)^(k-1) × e^(-(x/λ)^k) for x ≥ 0, and 0 for x < 0. The cumulative distribution function (CDF) is F(x) = 1 - e^(-(x/λ)^k) for x ≥ 0. The quantile function is Q(p) = λ × (-ln(1-p))^(1/k), where 0 ≤ p < 1.
How is the Weibull distribution related to the exponential distribution?
The exponential distribution is a special case of the Weibull distribution when the shape parameter k = 1. When k = 1, the Weibull PDF simplifies to f(x) = (1/λ) × e^(-x/λ), which is the exponential distribution with rate parameter 1/λ. Similarly, when k = 2, the Weibull distribution becomes the Rayleigh distribution.
What does the scale parameter λ represent in the Weibull distribution?
The scale parameter λ stretches or compresses the Weibull distribution along the x-axis. A larger λ spreads the distribution further to the right, while a smaller λ concentrates it closer to zero. Approximately 63.2% of the population lies below the scale parameter value, as F(λ) = 1 - e^(-1) ≈ 0.632. Thus λ is also called the characteristic life in reliability contexts.