Course in Probability Theory

3 .3 INDEPENDENCE 1 57Corollary . If {X j , 1 < j < n} are independent r .v .'s with finite expectations,then(6) c fJX j = 11 c'(X j) .nnj=1 j=1This follows at once by induction fromtwo r .v .'sftx1and11 Xjj=1 j=k+1n(5), provided we observe that theare independent for each k, I < k < n - 1 . A rigorous proof of this fact maybe supplied by Theorem 3 .3 .2 .Do independent random variables exist? Here we can take the cue fromthe intuitive background of probability theory which not only has given risehistorically to this branch of mathematical discipline, but remains a source ofinspiration, inculcating a way of thinking peculiar to the discipline . It maybe said that no one could have learned the subject properly without acquiringsome feeling for the intuitive content of the concept of stochastic independence,and through it, certain degrees of dependence . Briefly then : events aredetermined by the outcomes of random trials . If an unbiased coin is tossedand the two possible outcomes are recorded as 0 and 1, this is an r.v ., and ittakes these two values with roughly the probabilities 1 each . Repeated tossingwill produce a sequence of outcomes . If now a die is cast, the outcome maybe similarly represented by an r.v . taking the six values 1 to 6 ; again thismay be repeated to produce a sequence . Next we may draw a card from apack or a ball from an urn, or take a measurement of a physical quantitysampled from a given population, or make an observation of some fortuitousnatural phenomenon, the outcomes in the last two cases being r .v .'s takingsome rational values in terms of certain units ; and so on . Now it is veryeasy to conceive of undertaking these various trials under conditions suchthat their respective outcomes do not appreciably affect each other ; indeed itwould take more imagination to conceive the opposite! In this circumstance,idealized, the trials are carried out "independently of one another" and thecorresponding r .v .'s are "independent" according to definition . We have thus"constructed" sets of independent r .v.'s in varied contexts and with variousdistributions (although they are always discrete on realistic grounds), and thewhole process can be continued indefinitely .Can such a construction be made rigorous? We begin by an easy specialcase .