Barkley, Henry, and Bao (1996) examine patterns of spatial structural change for functional economic areas in North Carolina, South Carolina, and Georgia by estimating population density functions
using census tract data.
Nonparametric estimators are flexible, allowing the slope of density functions
to vary across the metropolitan area.
Recent papers propose methods where a hat function that approximates the respective probability density function
or probability vector is constructed (e.
The respective cumulative density functions
The method assumes that the errors follow a Gaussian probability density function
(PDF) with zero mean value and some known correlation matrix.
If the distribution G (x) has density g (x) the class will have a probability density function
(pdf) given by
Probability Density Function
(PDF) The Weibull's form of PDF is 
PNN is a parallel algorithm developing from Bayesian classification and the Parzen window probability density function
estimator (Umamaheswari, 2016; Shih, 2016).
Following Rice's  description of the probability density function
for load peaks, Bendat  showed how to calculate the fatigue damage for a narrowband random loading input.
From the families of multidimensional probability distributions described in literature ,  we chose the Gaussian, Dirichlet and gamma distributions for analysis because their probability density functions
are easy to use in analytical expressions.
A variety of strain energy density functions
have been extracted from Rivlin's model.
2](v) are the probability density functions
(pdfs) corresponding to the cdfs [F.