How this calculator works
This is the two-parameter Weibull distribution with location fixed at zero. Shape below one gives decreasing hazard, one gives constant hazard, and above one gives increasing hazard. At the scale value, cumulative probability is about 63.2% for every shape. Extremely small shapes can make moments exceed numeric range.
Formula
F(x) = 1-exp(-(x/λ)^k); f(x) = (k/λ)(x/λ)^(k-1)exp(-(x/λ)^k); E[X] = λΓ(1+1/k); Var[X] = λ²[Γ(1+2/k)-Γ(1+1/k)²].
Worked example
For shape 2, scale 3 and x = 3, cumulative probability is 1-exp(-1) = 0.6321206 and density is 2exp(-1)/3 = 0.2452530.
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Frequently asked questions
Is scale the average lifetime?
Only when shape is one. In general the mean uses the gamma function.
What if shape is one?
The model becomes exponential with rate 1/scale.
Can the density diverge at zero?
Yes, when shape is below one. The cumulative probability is still zero there.