1.6 Some common distributions

Definition Functions Measures


\(T\sim Exp( \lambda)\)

  • \(f(t)=\lambda exp(-\lambda t)\) where \(t \ge 0 and \lambda > 0\)

  • \(F(t)=1-exp(-\lambda t)\)

  • \(S(t)=exp(-\lambda t)\)

  • \(h(t) = \lambda\)

  • \(H(t) = \lambda t\)

\(E(T)=\int_0^{+\infty}uf(u) du= \frac{1}{\lambda}\)

\(Var(T)=E(T^2)-E(T)^2 = \ldots = \frac{1}{\lambda^2}\)


\(T\sim Weib(a,b)\) 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.