Weibull Distribution Probability Calculator | Six Values from k, λ, x
Enter shape k, scale λ and evaluation point x to read the Weibull PDF f(x), CDF F(x), mean, median, mode and standard deviation in one panel. Built for reliability engineering and wind-speed modeling.
💡 The shape parameter k drives the failure pattern
The Weibull distribution is a continuous probability distribution that models time-to-failure data and other non-negative quantities like wind speed. Its defining trait is that a single shape parameter k completely changes the curve.
When k<1 the failure rate decreases over time, which corresponds to the infant-mortality phase where manufacturing defects surface early. At k=1 the failure rate is constant, the distribution degenerates into the exponential distribution, and you have the random-failure phase. When k>1 the failure rate increases, signalling the wear-out phase. Lay those three regimes along a timeline and you get the bathtub curve that reliability engineers reason about.
The scale parameter λ stretches the curve horizontally and is also called the characteristic life: F(λ) always equals roughly 63.2%, so λ is the time by which about 63.2% of the population has failed.
This calculator accepts k from 0.05 to 20, λ from 0.01 to 1000 and x from 0 to 10000, clamping non-positive k or λ to the lower bound. Every edit recomputes all six values, so you can push the shape parameter around and watch the distribution change.
🧐 Frequently asked questions
Are k and β, or λ and η, the same thing? They are just different notation. Much of the reliability literature writes the shape parameter as β and the scale parameter as η (eta, the characteristic life). The k here maps to β and λ maps to η.
How is the mean computed? The mean is λ·Γ(1+1/k), where Γ is the gamma function. This tool evaluates it from a Lanczos log-gamma (lgamma) approximation using the Numerical Recipes coefficients. The standard deviation uses the same gamma function: √(λ²·(Γ(1+2/k) − Γ(1+1/k)²)).
Why does the mode show 0? The mode λ·((k−1)/k)^(1/k) is only defined for k>1. For k≤1 the peak of the density sits at x=0, so the mode is reported as 0.
Entering x=0 shows the PDF as "∞" For k<1, f(0) diverges, so it displays as ∞. At k=1 it is 1/λ, and for k>1 it is 0. The CDF is always 0 at x=0.
Does a large k approach a normal distribution? Around k≈3.5 the Weibull curve becomes nearly symmetric and visually close to a normal distribution, though it never becomes exactly normal.
📚 Why the bathtub curve matters
The bathtub curve splits a product's lifecycle into three regimes: infant mortality, random failures and wear-out. Each maps to k<1, k=1 and k>1 respectively. Setting a warranty window or scheduling preventive maintenance starts with figuring out which part of the curve a component sits in, and the shape parameter is exactly what tells you that.
Wind-energy analysts lean on the same distribution. The wind-speed distribution at a site is usually well described by a Weibull with a shape parameter near 2, and from the mean wind speed and that shape you can estimate the energy a turbine will produce. Long before spreadsheets, engineers plotted failure data on Weibull probability paper and read the shape parameter off the slope of the fitted line — a graphical method that still shapes how the distribution is taught.