50 Jahre Deutsche Statistische Gesellschaft Tagungen 1961

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XY, ) ( Y ] } . 76) can be extended to the case of several random variables. For instance, to calculate E { h ( X ,Y ,2))we apply the Law of Iterated Expectations twice as follows. 76) by conditioning on 2. Thus E { h , ( X ,Y ,Z)}= E { E { h ( X :Y ,2)I Z } } . 76) by conditioning on Y . As a result, we have E { h , ( X , Y , Z )I Z}= E { E [ h ( X , Y , Z 1)Y, Z]1 Z}. Therefore, we obtain E{1l(X,Y,2))= - w { E [ h ( X y, , Z)I y, ZI I 21). 77) provides a step-by-step calculation of an unconditional expectation involving more than one random variable.

B x ( ~. In other words, it is the smallest information structure containing B x ( o ) , , . . , B x ( ~ We ) . denote this information structure as at. It is easy to see that the sequence t = 0,1, . , BO c B1 Bt ' . ' 2 3. d,~,, at, c " ' c Thus, Dt, t = 0, 1, . . , forms a time-dependent information structure or a filtration on (n,3,P ) , called the natural information structure or the natural filtration' with respect to X ( t ) , t = 0 , 1 , . ' . Thus, if a sequence of random variables X ( t ) , t = 0,1, .

N,, is also multinomially distributed. For example, N1, AT2 is trinomially distributed with parameters n , q1 and q2. The random variables N1, N 2 , . , N,, are negatively correlated. In fact, + + + Gov(Nt,N J = -n4243, i # j. 39) It is easy to derive the moment generating function of N1, N2, . ,. ,z,) = [qlc" q2cz2 . . + qmcZmIn. f(X,Y) = 1 27r01 crz Jcq7 _ _2 (_, ! 41) for --co < z,y < 03. 24)thatX N ( p 1 , o f ) a n d Y N ( p 2 , a ; ) . Further, C o r r ( X ,Y ) = p. Thus, all the parameters are meaningful: the p's and cr's are the means and standard deviations of the random variables, and p is the correlation coefficient.

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