## 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.