Weibull Distribution Calculator
The Weibull distribution is the standard tool of reliability and survival analysis, able to represent improving, constant, or worsening failure rates through a single shape parameter. Enter the shape k, the scale lambda, and an evaluation point x. The calculator returns the probability density at x, the cumulative probability that failure has occurred by x, and the distribution's mean and variance computed from the gamma function. It models component lifetimes, wind speed distributions, and time-to-event data. The cumulative function is closed form, and the moments use a standard numerical gamma function.
Weibull distribution formulas
F(x) = 1 - e^(-(x/lambda)^k) for x >= 0, else 0
f(x) = (k/lambda) * (x/lambda)^(k-1) * e^(-(x/lambda)^k) for x >= 0
Mean = lambda * gamma(1 + 1/k)
Variance = lambda^2 * [gamma(1 + 2/k) - gamma(1 + 1/k)^2]
The shape k sets the failure pattern and the scale lambda sets the characteristic life, the point at which about 63.2 percent of items have failed.
Worked example and notes
- With k = 2, lambda = 1, the distribution is a Rayleigh distribution.
- At x = lambda the cumulative probability is always 1 minus e^(-1), about 0.6321, for any k.
- The mean for k = 2, lambda = 1 is gamma(1.5), about 0.8862.
- k below 1 models early-life failures; k above 1 models wear-out failures.
- k = 1 reduces the Weibull to the exponential distribution.
Weibull distribution: frequently asked questions
What is the Weibull distribution?
The Weibull distribution is a flexible continuous distribution on the positive reals defined by a shape parameter k and a scale parameter lambda. It is the workhorse of reliability engineering because its shape parameter can model decreasing, constant, or increasing failure rates over time.
What does the shape parameter k mean?
When k is below 1 the failure rate decreases over time (infant mortality). When k equals 1 the rate is constant and the distribution is exponential. When k is above 1 the failure rate increases over time (wear-out). At k near 3.4 the shape is close to a normal curve.
What are the density and cumulative function?
For x at or above 0 the cumulative distribution function is 1 minus e to the power minus (x over lambda) to the power k. The density is (k over lambda) times (x over lambda) to the power (k minus 1) times that same exponential factor.
What are the mean and variance?
The mean is lambda times the gamma function of (1 plus 1 over k). The variance is lambda squared times the gamma function of (1 plus 2 over k) minus the square of the gamma function of (1 plus 1 over k). The calculator evaluates the gamma function numerically.
What are valid inputs?
The shape k and scale lambda must both be strictly positive. The evaluation point x can be any real number; for x at or below 0 the density and cumulative value are 0, since the Weibull distribution is supported on the non-negative reals.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Weibull distribution.
- NIST Digital Library of Mathematical Functions: Gamma function.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.