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

2.10 INDEPENDENCE

Independence in probability is an important property that is often assumed when modeling an experiment, which usually leads to simplified analyses. Intuitively, we can say that two events are independent if they have no influence on the outcomes of each other. It is defined as follows.

Definition: Independent Events Events A and B are independent if and only if

(2.62)

Probability P(AB) is known as the joint probability of A and B (i.e., they occur jointly). The individual probabilities P(A) and P(B) in the context of a joint probability are known as marginal probabilities. The definition of independence is clear for the following simple example.

Example 2.41. The probability of observing two heads when simultaneously tossing two fair coins is P(HH) = P(H)P(H) = (1/2)(1/2) = 1/4. It is obvious from a physical view of the dynamics of tossing a coin, there is no reason why the outcome of one coin should influence the other. Intuitively, and from the frequency interpretation of probability, the probabilities should split as in (2.62) for independent events. Likewise, the probability of any pair of outcomes when tossing two fair dice is where .

Observe from the definition ...