Random-Variable Space

The purpose of this chapter is to give an example of a geometric algebra over a space other than a real vector space. From Axiom 3.1 of geometric algebra, it follows that a geometric algebra can also be formed over a nite-dimensional Hilbert space. The particular example considered here is the Hilbert space of random variables. Another example could be the Hilbert space of the basis functions of a nite Fourier series. In all cases, the concepts of blades, null spaces, intersections, and combinations of subspaces are still valid in the geometric algebra over the Hilbert space, even though they may not have the same geometric meaning. However, a geometric meaning may be given to otherwise abstract operations in this way. This may help us to gain additional insights into various elds and to draw parallels between different fields.

While the Hilbert space of random variables is well known, the geometric algebra over this Hilbert space has not so far been treated. The main results of this chapter are about how the variance, the co-variance, and the the Cauchy{Schwarz inequality follow directly from operations on blades of random variables. Furthermore, an equation for the correlation coeffcient between an arbitrary number of random variables is derived.

The plan of this chapter is as follows. In Sect. 10.1, some basic properties of random variables are introduced and operations between random variables and their corresponding density functions are dened. In Sect. 10.2, the Hilbert space of random variables is developed using the previously dened notation. Here, also, the Dirac delta random variable is dened, and it is shown that it can be interpreted as the “direction” of the expectation value of a random variable. In this way, a homogeneous random-variable Hilbert space can be developed. After this initial work, the geometric algebra over the homogeneous random-variable space is introduced in Sect. 10.3. Note that Sects. 10.1 and 10.2 give a formal treatment of the Hilbert space of random variables. Readers who are interested mainly in the treatment of the geometric algebra over the Hilbert space of random variables may go straight to Sect. 10.3, with a quick look at Sect. 10.2.3.