Weibull Distribution Calculator

The Weibull distribution is a continuous probability distribution defined by two primary parameters: the shape parameter (k > 0) and the scale parameter (lambda > 0). The probability density function (PDF) is expressed as f(x) = (k / lambda) * (x / lambda)^(k - 1) * exp(-(x / lambda)^k) for x >= 0. The cumulative distribution function (CDF) calculates the probability that a variable falls below a threshold x, given by F(x) = 1 - exp(-(x / lambda)^k). Derived metrics such as mean, variance, standard deviation, and median utilize mathematical gamma functions tailored to the specific shape and scale inputs.

Consider a mechanical component reliability test where the scale parameter (lambda) is set to 100 hours and the shape parameter (k) is 2.0, representing an increasing failure rate over time. To find the cumulative probability of failure by x = 50 hours, we substitute these values into the CDF formula: F(50) = 1 - exp(-(50 / 100)^2). This simplifies to 1 - exp(-0.5^2) = 1 - exp(-0.25). Evaluating exp(-0.25) yields approximately 0.7788, resulting in a cumulative failure probability of 1 - 0.7788 = 0.2212, or roughly 22.12%. Similarly, the probability density function at this exact point evaluates the instantaneous failure likelihood, while the mean and variance computations outline the expected operational lifespan and its statistical spread.

Ensure your scale parameter is always greater than zero and representative of the characteristic life of your dataset. Pay close attention to the shape parameter: a shape under 1 indicates a decreasing failure rate (infant mortality), a shape of 1 reflects a constant failure rate, and a shape greater than 1 signifies wear-out failures. Always verify units of measurement for your input variables, such as hours, cycles, or stress levels, to maintain calculation consistency.