1.6 Some common distributions

Definition	Functions	Measures
Exponential \(T\sim Exp( \lambda)\)	\(f(t)=\lambda exp(-\lambda t)\) where \(t \ge 0 and \lambda > 0\) \(F(t)=1-exp(-\lambda t)\) \(S(t)=exp(-\lambda t)\) \(h(t) = \lambda\) \(H(t) = \lambda t\)	\(E(T)=\int_0^{+\infty}uf(u) du= \frac{1}{\lambda}\) \(Var(T)=E(T^2)-E(T)^2 = \ldots = \frac{1}{\lambda^2}\)
Weibull \(T\sim Weib(a,b)\) with \(a\) shape and \(b\) scale	\(f(t)=\frac{a}{b} (\frac{t}{b})^{a-1} exp^{-\left(\frac{t} {b} \right)^a}\) where \(t\ge 0\) and \(a,b> 0\) \(F(t)= 1-exp^{- \left(\frac{t}{b} \right)^a}\) \(S(t)=exp^{-\left( \frac{t}{b} \right)^a}\) \(h(t)=ab^{-a}t^{a-1}\) \(H(t)=(\frac{t} {b})^a\)	\(E(T)=b\Gamma \left(1+ \frac{1}{a}\right)\) \(Var(T) = b^2 \Gamma \left(1+ \frac{2}{a}\right) - b^2 \left [ \Gamma \left(1+ \frac{1}{a}\right)\right]^2\) where, \(\Gamma(k)\) is the gamma function. \(\Gamma (k) = \int_0^{+\infty} u^{k-1} exp^{-u}du\)

There are other distributions such as Log-Normal, Log-Logistic, Pareto, Rayleigh, Gomptertz, or even more. For more details see http://data.princeton.edu/pop509/ParametricSurvival.pdf.