Probability Density Function - Families of Densities

Families of Densities

It is common for probability density functions (and probability mass functions) to be parametrized, i.e. containing unspecified parameters. For example, the normal distribution is normally parametrized in terms of the mean and the variance, denoted by and respectively, giving the family of densities

It is important to keep in mind the difference between the domain of a family of densities and the parameters of the family. Different values of the parameters describe different distributions on the same sample space; this sample space is the domain of the family. A given set of parameters describes a single distribution, and the sample space is the domain of the actual random variable that this distribution describes. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables with values in the domain, are part of the normalization factor of a distribution and outside the kernel of the distribution. Since the parameters are constants, reparameterizing a family of densities in terms of different parameters means simply substituting the new parameters into the formula in the obvious way. Changing the domain of a probability density, however, is trickier and requires more work: see the section below on change of variables.