Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

2.13 DISCRETE SAMPLE SPACES

For discrete sample spaces (finite or countably infinite), it is straightforward to assign probabilities because events are expressed in terms of the individual sample points in Ω, which in turn are assigned probabilities satisfying the axioms of probability. As mentioned earlier, we generally assume for discrete Ω.

Example 2.46. Consider an experiment where the countable outcomes {xn} are the natural numbers . Suppose we assign probabilities to these outcomes as follows:

(2.89)

where 0<p<1. This probability assignment yields the geometric random variable described in Chapter 3. In fact, it corresponds to the experiment of successively tossing a single coin with P(H) = p until the first H appears. Even though an infinite number of outcomes are possible, these probabilities sum to one:

(2.90)

where the closed-form expression for an infinite series in Appendix E has been used. This is an example of a convergent series that has been scaled by p so that it converges to 1, and is due to the fact that there is a countable number of terms. In fact, the terms of ...