Random Variable - Random Signals and Processes Primer with MATLAB - page 30

Every point from the space S must map to only one point on the real axis.
This means that it is not possible to map the particular outcome s i of the space S to
two or more points on a real axis.
However, it is possible to map different outcomes s i to the same point at x line.
Therefore, the mapping can be one-to-one , in which every outcome s i is mapped
onto only one point or many-to-one , in which different outcomes are mapped onto
the same point, as shown in Fig. 2.1 .
The numerical values x on the real axis are the range of the random variable.
We will denote the random variable with capital letters, such as X , Y , Z , and the
corresponding range with lowercase letters, such as x , y , and z . If the range of
the random variable is discrete, then the random variable is discrete , as shown in
Fig. 2.1 . Otherwise it is a continuous random variable (see Fig. 2.2 ). Let us recall
that a discrete set is a countable set of points that can also be infinite. A continuous
set, on the other hand, is always uncountable and infinite.
Fig. 2.1 Concept of a discrete random variable. ( a ) one-to-one mapping ( b ) many-to-one mapping
Fig. 2.2 Concept of a continuous random variable