By Lloyd Fisher, Z. W. Birnbaum, E. Lukacs, John N. McDonald

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The joint moment generating function (if it exists) is M(tl9... ,tk) = £(exp£? =1 t/Xj) (tf a drdimensional vector). (a) Assume that the joint moment generating function exists. 4) that the moment generating function determines the joint distribution. (b) Show (in the case where each Xt has a probability density function) that the X, are independent if and only if M(tl9... 9tk) = M(ti909... ,0)M(0,t 2 ,0,... ,0) · · · M(0,... ,tk). ) [Hint: Show that M(tl909... 9 (a) (BesseVs inequality) Let Pi9 i = 1, .

Birnbaum, and L. Fisher, Pocketbook of Statistical Tables. Dekker, New York, 1977. 4 THE k-SAMPLE COMPARISON OF MEANS (ONE- WA Y ANAL YSIS OF VARIANCE) Let us slightly generalize the problem of the previous chapter. Suppose that instead of two independent random samples we now have independent random samples from k(k>2) populations. Suppose we model this situation as follows: Xij ~ Ν{μί9σ2\ i = 1,... ,/c, j = 1,... ,nh all independent random variables; that is, the ith population is sampled nf times, and it has also been assumed that the variance is the same in each population.

8 (a) If b l 5 . . ,b„ is an orthonormal basis for Rn9 then any X is uniquely expressed in terms of this basis as x = x1b1 +--- + ΧΛ, where Xt = \%. 1. Find the matrix P relating the bases in terms of the angle Θ. b2 f22. 70 If C"xn is a real, symmetric matrix and P an orthogonal matrix such that PCP' = D, where D is diagonal, show that the diagonal elements of D are the eigenvalues of C. 77 Find an orthogonal matrix P 2 x 2 such that P(? l)P' is diagonal. 72 Show that a real symmetric C is positive (nonnegative) definite if and only if all the eigenvalues are positive (nonnegative).