Karl Pearson showed that if we know the first four moments of a distribution, we can construct a density function that is consistent with those moments. This can provide a neat way to build density functions that approximate a given set of data. For instance, for a given data set, let us suppose that:
![[Graphics:Images/index_gr_1.gif]](Images/index_gr_1.gif)
denoting estimates of the mean, and of the second, third and fourth central moments. The Pearson family consists of 7 main Types, so our first task is to find out which type this data is consistent with. We do this with mathStatica's PearsonPlot function:
![[Graphics:Images/index_gr_2.gif]](Images/index_gr_2.gif)
Fig. 1: The chart for the Pearson system
The big black dot in Fig. 1 is in the Type I zone. Then, the fitted Pearson density and its domain are immediately given by:
![[Graphics:Images/index_gr_7.gif]](Images/index_gr_7.gif)
The actual data used to create this example is grouped data depicting the number of sick people (freq) at different ages (X):
![[Graphics:Images/index_gr_9.gif]](Images/index_gr_9.gif)
We can easily compare the histogram of the empirical data with our fitted Pearson pdf:
![[Graphics:Images/index_gr_10.gif]](Images/index_gr_10.gif)
Fig. 2: The data histogram and the fitted Pearson pdf