Editing Probability density function (section)

== Families of densities ==
It is common for probability density functions (and [[probability mass function]]s) to be parametrized—that is, to be characterized by unspecified [[parameter]]s. For example, the [[normal distribution]] is parametrized in terms of the [[mean]] and the [[variance]], denoted by <math>\mu</math> and <math>\sigma^2</math> respectively, giving the family of densities
<math display="block">
   f(x;\mu,\sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }.
</math>
Different values of the parameters describe different distributions of different [[random variable]]s on the same [[sample space]] (the same set of all possible values of the variable); this sample space is the domain of the family of random variables that this family of distributions describes.  A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the [[normalization factor]] of a distribution (the multiplicative factor that ensures that the area under the density—the probability of ''something'' in the domain occurring— equals 1). This normalization factor is outside the [[kernel (statistics)|kernel]] of the distribution.

Since the parameters are constants, reparametrizing a density in terms of different parameters to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones.