C.1. Probability TheoryLet's first consider a random experiment, such as producing random logic 0s and 1s. The sample space of the experiment, usually denoted by the symbol S , consists of the set of all possible outcomes , indicated by ‰ . In this case, for integers 0 and 1, the sample space is S ={0, 1}. We define an event , A, to be a subset of sample space, which may consist of any number of sample points. Thus, if define event A ={1}, the event consists of only one point. The union of two events, A and B, covers all outcomes belonging to both events and is shown by A B. The intersection of events A and B refers to all outcomes shared between the two events and is shown by A B. The complement of event A is an event that covers all events but the ones belonging to A and is shown by ‚. The probability of an event A is shown by P [A] and always 0 P [A] 1. Also, if A B = 0, then: Equation C.1
In most engineering cases, especially in analyzing random signals and systems, we are often interested in the conditional probability. The probability that event A occurs given that event B has occurred is defined as Equation C.2
Two events A and B are independent of each other if Equation C.3
C.1.1. Bernulli and Binomial Sequential LawsTwo fundamental laws of sequential experiments are Bernulli and binomial trials. A Bernulli trial is a sequence of repeated independent random experiments whose outcomes are either a success with probability p or a failure with probability 1 - p . A binomial trial measures k successes, each with probability p in n independent Bernulli trials. As a result, the probability of having k successes in n trials is Equation C.4
where C.1.2. Counting and Sampling MethodsOne can simply observe that the probability of an event A can also be computed through a counting method by: Equation C.5
where n i is the number of outcomes in event A, and n is the total number of outcomes in global sample space. We note that the larger n i and n are, the more accurate the probability of A becomes. |