Parametric Families of Density Functions

Abstract

Some specific functional forms of probability density functions that have been found to be useful in statistical applications are examined in this chapter. The selection includes a number of the more commonly used densities.¹ Our density function definitions will actually identify parametric families of density functions. That is, the algebraic expressions for the density functions will contain one or more unknowns, called parameters, which can be assigned values chosen from a set of admissible values called the parameter space. A specific member of a family of densities will be associated with each specific value of the parameters contained in the parameter space. The general notation f(x; θ) will be used to distinguish elements in the domain of the density function from elements in the parameter space of the parametric family of functions. In particular, the argument preceding the semicolon (x in the present case) represents domain elements, while the argument following the semicolon (θ in this case) represents elements of the parameter space.