Weibull distribution

The two-parameter Weibull distribution is a positive lifetime model with shape $k>0$ and scale $\lambda >0$. Its Probability density function and Survival function areIts Cumulative distribution function is $1-S(y)$. The Hazard function is $h(y)=k{\lambda }^{-1}(y/\lambda {)}^{k-1}$: it decreases for $k<1$, is constant for $k=1$, and increases for $k>1$. This permits decreasing or increasing failure rates within one family.^[ref]

The mean is $\lambda \Gamma (1+1/k)$ and the variance is ${\lambda }^2[\Gamma (1+2/k)-\Gamma (1+1/k{)}^2]$. The substitution $v=(y/\lambda {)}^k$ reduces the power-moment integrals to gamma integrals. At $k=1$ the law is exponential with rate $1/\lambda$; at $k=2$ its survival is $e^{-(y/\lambda {)}^2}$.

Inverse transform sampling is explicit: for $U$ uniform on $(0,1)$, take $Y=\lambda (-\log U{)}^{1/k}$. Conditional survival after an already attained age $a$ is $S(a+u)/S(a)$, which depends on $a$ unless $k=1$. This is why an Unknown entry age in a Weibull survival segment affects the likelihood. The scale parameter here is $\lambda$; inverse-scale parameterizations instead use its reciprocal.