| Weibull distribution | CDF | F ( x ) = { 1 − e − ( x / λ ) k , x ≥ 0 , 0 , x < 0. {\displaystyle F(x)={\begin{cases}1-e^{-(x/\lambda )^{k}},&x\geq 0,\\0,&x<0.\end{cases}}} | 1.00 | infobox |
| Weibull distribution | CF | ∑ n = 0 ∞ ( i t ) n λ n n ! Γ ( 1 + n / k ) {\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k)} | 1.00 | infobox |
| Weibull distribution | Entropy | γ ( 1 − 1 / k ) + ln ( λ / k ) + 1 {\displaystyle \gamma (1-1/k)+\ln(\lambda /k)+1\,} | 1.00 | infobox |
| Weibull distribution | Excess kurtosis | (see text) | 1.00 | infobox |
| Weibull distribution | Kullback–Leibler divergence | see below | 1.00 | infobox |
| Weibull distribution | Mean | λ Γ ( 1 + 1 / k ) {\displaystyle \lambda \,\Gamma (1+1/k)\,} | 1.00 | infobox |
| Weibull distribution | Median | λ ( ln 2 ) 1 / k {\displaystyle \lambda (\ln 2)^{1/k}\,} | 1.00 | infobox |
| Weibull distribution | MGF | ∑ n = 0 ∞ t n λ n n ! Γ ( 1 + n / k ) , k ≥ 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {t^{n}\lambda ^{n}}{n!}}\Gamma (1+n/k),\ k\geq 1} | 1.00 | infobox |
| Weibull distribution | Mode | { λ ( k − 1 k ) 1 / k , k > 1 , 0 , k ≤ 1. {\displaystyle {\begin{cases}\lambda \left({\frac {k-1}{k}}\right)^{1/k}\,,&k>1,\\0,&k\leq 1.\end{cases}}} | 1.00 | infobox |
| Weibull distribution | Parameters | λ ∈ ( 0 , + ∞ ) {\displaystyle \lambda \in (0,+\infty )\,} scale k ∈ ( 0 , + ∞ ) {\displaystyle k\in (0,+\infty )\,} shape | 1.00 | infobox |
| Weibull distribution | PDF | f ( x ) = { k λ ( x λ ) k − 1 e − ( x / λ ) k , x ≥ 0 , 0 , x < 0. {\displaystyle f(x)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda… | 1.00 | infobox |
| Weibull distribution | Quantile | Q ( p ) = λ ( − ln ( 1 − p ) ) 1 k {\displaystyle Q(p)=\lambda (-\ln(1-p))^{\frac {1}{k}}} | 1.00 | infobox |
| Weibull distribution | Skewness | Γ ( 1 + 3 / k ) λ 3 − 3 μ σ 2 − μ 3 σ 3 {\displaystyle {\frac {\Gamma (1+3/k)\lambda ^{3}-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}} | 1.00 | infobox |
| Weibull distribution | Support | x ∈ [ 0 , + ∞ ) {\displaystyle x\in [0,+\infty )\,} | 1.00 | infobox |
| Weibull distribution | Variance | λ 2 [ Γ ( 1 + 2 k ) − ( Γ ( 1 + 1 k ) ) 2 ] {\displaystyle \lambda ^{2}\left[\Gamma \left(1+{\frac {2}{k}}\right)-\left(\Gamma \left(1+{\frac {1}{k}}\right)\right)^{2}\right]\,} | 1.00 | infobox |
| Weibull distribution | is a | hyperbolastic distribution of type III.Cumulative distribution functionThe cumulative distribution function for the Weibull distribution is F | 0.90 | text |
| Weibull distribution | is a | maximum entropy distribution for a non-negative real random variate with a fixed expected value of xk equal to λk and a fixed expected value of ln | 0.90 | text |
| Weibull distribution | is a | hyperbolastic distribution of type III | 0.90 | text |
| Weibull distribution | is a | generalized gamma distribution with both shape parameters equal to k.The translated Weibull distribution | 0.90 | text |
| Weibull distribution | is a | special case of the generalized extreme value distribution | 0.90 | text |
