Piecewise Distributions

Some density functions take a bipartite form. To illustrate, let us suppose [Graphics:Images/index_gr_1.gif] is a continuous random variable, [Graphics:Images/index_gr_2.gif], with pdf

[Graphics:Images/index_gr_3.gif]

where [Graphics:Images/index_gr_4.gif]. We enter this as:

[Graphics:Images/index_gr_5.gif]

This is known as the Inverse Triangular distribution, as is clear from a plot of [Graphics:Images/index_gr_6.gif], as illustrated in Fig.Ūd1.

[Graphics:Images/index_gr_7.gif]

[Graphics:Images/index_gr_8.gif]

Fig. 1: The Inverse Triangular pdf, when [Graphics:Images/index_gr_9.gif](bold), [Graphics:Images/index_gr_10.gif](plain), [Graphics:Images/index_gr_11.gif](dashed)

Here is the cdf, [Graphics:Images/index_gr_12.gif]:

[Graphics:Images/index_gr_13.gif]
[Graphics:Images/index_gr_14.gif]

Note that the solution depends on whether [Graphics:Images/index_gr_15.gif] or [Graphics:Images/index_gr_16.gif]. Figure 2 plots the cdf at the same three values of [Graphics:Images/index_gr_17.gif] used in Fig.1.

[Graphics:Images/index_gr_18.gif]

Fig. 2: The Inverse Triangular cdf, when [Graphics:Images/index_gr_19.gif](bold), [Graphics:Images/index_gr_20.gif](plain), [Graphics:Images/index_gr_21.gif](dashed)

mathStatica operates on bipartite distributions in the standard way. For instance, the mean [Graphics:Images/index_gr_22.gif] is given by:

[Graphics:Images/index_gr_23.gif]
[Graphics:Images/index_gr_24.gif]

while the entropy is given by [Graphics:Images/index_gr_25.gif]:

[Graphics:Images/index_gr_26.gif]
[Graphics:Images/index_gr_27.gif]