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Let fY i (y i ) be the marginal pdf or pmf of Y i . , yn ) = fY 1 (y 1 ) · · · fY n (y n ) = fY i (y i ). 7. 18. , n. , n. Then the random vectors X i = hi (Y i ) are independent. There are three important special cases. i) If pji = 1 so that each hi is a real valued function, then the random variables Xi = hi (Y i ) are independent. , Xn are independent. X. , Xm ) and assume that Y h(Y ) is a vector valued function of Y alone and if g(X) is a vector valued function of X alone, then h(Y ) and g(X) are independent random vectors.

Yn are independent with mgfs mYi (t). Then the mgf of W = ni=1 (ai + biYi ) is n mW (t) = exp(t n ai ) i=1 mYi (bi t). 23) i=1 Proof of g): Recall that exp(w) = ew and exp( nj=1 dj ) = nj=1 exp(dj ). It can be shown that for the purposes of this proof, that the complex constant i in the characteristic function (cf) can be treated in the same way as if it were a real constant. Now n φW (t) = E(e itW ) = E(exp[it (aj + bj Yj )]) j=1 CHAPTER 2. 5 the expected value of a product of independent random variables is the product of the expected values of the independent random variables.

Let W = ni=1 Yi . Then n n n a) E(E) = E( i=1 Yi ) = i=1 E(Yi ) = i=1 µi , and b) V (W ) = V ( ni=1 Yi ) = ni=1 V (Yi ) = ni=1 σi2 . A statistic is a function of the random sample and known constants. A statistic is a random variable and the sampling distribution of a statistic n is the distribution of the statistic. , an are constants. The following theorem n shows how to find the mgf and characteristic function of such statistics. 16. a) The characteristic function uniquely determines the distribution.

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