### Basics: Working with arbitrary distributions

**mathStatica** adds about 100 new functions to *Mathematica*. But most of the time, we can get by with just four of them:

**Table 1: **Core functions for a random variable with density

This ability to handle plotting, expectations, probability, and transformations, with just four functions, makes the **mathStatica** system very easy to use, even for those not familiar with *Mathematica*.

To illustrate, let us suppose the continuous random variable has probability density function (pdf)

, for .
We enter this as:

This is known as the Arc--Sine distribution. Here is a plot of :

**Fig. 1: ** The Arc-Sine pdf

Here is the cumulative distribution function (cdf), , which also provides the clue to the naming of this distribution:

The mean, , is:

while the variance of is:

The moment of is :

Now consider the transformation to a new random variable such that . By using the Transform and TransformExtremum functions, the pdf of , say , and the domain of its support can be found:

So, we have started out with a quite arbitrary pdf , transformed it to a new one , and since both density *g* and its domain have been entered into *Mathematica*, we can also apply the **mathStatica** tool set to density . For example, use PlotDensity[*g*] to plot the pdf of .