Parameterised distributions Suppose that we have an
indexed family of distributions. If the index is also a parameter of the members of the family, then the family is a
parameterized family. Among
parameterized families of distributions are the
normal distributions, the
Poisson distributions, the
binomial distributions, and the
exponential family of distributions. For example, the family of
normal distributions has two parameters, the
mean and the
variance: if those are specified, the distribution is known exactly. The family of
chi-squared distributions can be indexed by the number of
degrees of freedom: the number of degrees of freedom is a parameter for the distributions, and so the family is thereby parameterized.
Measurement of parameters In
statistical inference, parameters are sometimes taken to be unobservable, and in this case the statistician's task is to estimate or infer what they can about the parameter based on a
random sample of observations taken from the full population. Estimators of a set of parameters of a specific distribution are often measured for a population, under the assumption that the population is (at least approximately) distributed according to that specific probability distribution. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom in a
Pearson's chi-squared test). Even if a family of distributions is not specified, quantities such as the
mean and
variance can generally still be regarded as statistical parameters of the population, and statistical procedures can still attempt to make inferences about such population parameters.
Types of parameters Parameters are given names appropriate to their roles, including the following: •
location parameter •
dispersion parameter or
scale parameter •
shape parameter Where a probability distribution has a domain over a set of objects that are themselves probability distributions, the term
concentration parameter is used for quantities that index how variable the outcomes would be. Quantities such as
regression coefficients are statistical parameters in the above sense because they index the family of
conditional probability distributions that describe how the
dependent variables are related to the independent variables. ==Examples==