INTRODUCTION TO STATISTICS AND ECONOMETRICS
Statistical Independence
We shall first define the concept of statistical (stochastic) independence for a pair of events. Henceforth it will be referred to simply as “independence.”
definition 2.4.1 Events A and В are said to be independent if P(A) = P(A I B).
The term “independence” has a clear intuitive meaning. It means that the probability of occurrence of A is not affected by whether or not В has occurred. Because of Theorem 2.4.1, the above equality is equivalent to P(A)P(B) = P(A П B) or to P(B) = P(B I A).
Since the outcome of the second toss of a coin can be reasonably assumed to be independent of the outcome of the first, the above formula enables us to calculate the probability of obtaining heads twice in a row when tossing a fair coin to be У4.
Definition 2.4.1 needs to be generalized in order to define the mutual independence of three or more events. First we shall ask what we mean by the mutual independence of three events, A, B, and C. Clearly we mean pairwise independence, that is, independence in the sense of Definition 2.4.1 between any pair. But that is not enough. We do not want A and В put together to influence C, which may be stated as the independence between А П В and C, that is, P(A П В) = P(A П В C). Thus we should have
Р(АПБПС) = P[(A П В) I C]P(C) = P(A П B)P(C)
Note that independence between А П C and В or between В П C and A follows from the above. To summarize,
DEFINITION 2.4.2 Events A, B, and C are said to be mutually independent if the following equalities hold:
(2.4.5) P(A П B) = P(A)P(B).
(2.4.6) P(A П C) = P(A)P(C).
(2.4.7) P(B П C) = P(B)P(C).
(2.4.8) P(A П В П C) = P(A)P(B)P(C).
We can now recursively define mutual independence for any number of events:
definition 2.4.3 Events Аь A2, . . . , An are said to be mutually independent if any proper subset of the events are mutually independent and
P(Ai П A2 П . . . П An) = J°(Aj)P(A2) • • • P{An).
The following example shows that pairwise independence does not imply mutual independence. Let A be the event that a head appears in the first toss of a coin, let В be the event that a head appears in the second toss, and let C be the event that either both tosses yield heads or both tosses yield tails. Then A, B, and C are pairwise independent but not mutually independent, because P(A П В П С) = P(A П B) = %, whereas P(A)P(B)P(C) =%.
