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

2.5 SUMMARY OF A RANDOM EXPERIMENT

It will be useful at this point to summarize our description of a random experiment for which we want to assign a probability measure:

• The sample space Ω is the collection of all outcomes of an experiment. Depending on the type of experiment, it may contain a finite, countably infinite, or uncountable number of elements.
• An event of an experiment is a subset of Ω consisting of one or more outcomes, and which usually share some feature. Since the goal is to assign a probability measure to events that is consistent, we may not want to include all possible subsets of Ω.
• A σ-field is a collection of subsets of Ω that satisfy the algebraic conditions in (i) and , as well as (iii), (iv), and . One can envision constructing a σ-field by starting with some subset of Ω and expanding the number of subsets until the collection satisfies (i) and .
• If Ω is finite or countably infinite, then all subsets described by the power set comprise a useful σ-field for the ...