• The
Poisson family of distributions is parametrized by a single number : : \mathcal{P} = \Big\{\ p_\lambda(j) = \tfrac{\lambda^j}{j!}e^{-\lambda},\ j=0,1,2,3,\dots \ \Big|\;\; \lambda>0 \ \Big\}, where is the
probability mass function. This family is an
exponential family. • The
normal family is parametrized by , where is a location parameter and is a scale parameter: : \mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{1}{\sqrt{2\pi}\sigma} \exp\left(-\tfrac{(x-\mu)^2}{2\sigma^2}\right)\ \Big|\;\; \mu\in\mathbb{R}, \sigma>0 \ \Big\}. This
parametrized family is both an
exponential family and a
location-scale family. • The
Weibull translation model has a three-dimensional parameter : : \mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{\beta}{\lambda} \left(\tfrac{x-\mu}{\lambda}\right)^{\beta-1}\! \exp\!\big(\!-\!\big(\tfrac{x-\mu}{\lambda}\big)^\beta \big)\, \mathbf{1}_{\{x>\mu\}} \ \Big|\;\; \lambda>0,\, \beta>0,\, \mu\in\mathbb{R} \ \Big\}, where \beta is the
shape parameter, \lambda is the
scale parameter and \mu is the
location parameter. • The
binomial model is parametrized by , where is a non-negative integer and is a probability (i.e. and ): : \mathcal{P} = \Big\{\ p_\theta(k) = \tfrac{n!}{k!(n-k)!}\, p^k (1-p)^{n-k},\ k=0,1,2,\dots, n \ \Big|\;\; n\in\mathbb{Z}_{\ge 0},\, p \ge 0 \land p \le 1\Big\}. This example illustrates the definition for a model with some discrete parameters. ==General remarks==