114 PROBLEMS AND SOLUTIONS

Prove

42 dz dOo sin OoP’(Oo, 0o, Z, (R), )PT;’(Oo, 0o, Z, 0’, ’) dOo0 0 0

0 for/el’ and/or mere’,

4e O’ O’ ( ’)] for 1’(2/+ 1)2 P[cos

cos + sin sin cos and

m m’,

Problem 70-20 corrected, A Minimization for Markov Chains, by PAUL J.SCHWWITZR (Institute for Defense Analyses).Consider

N N

min E EP j=

where the minimum is taken over all N x N single-chained stochastic matricesP and where r denotes the unique stationary distribution for P (i.e., P => 0,= Pit 1, z rP, =1 rri 1). Show that theminimum is 1/N and is achieveduniquely for

0

P*= 0

0

0 0 0 1

0 0 0 0

0 0

0 0

Problem 71-1, A Functional Equation, by L. A. SHEPP (Bell Telephone Laboratories).

Prove that there is only one nonnegative function F for which

F {f F(u) du} x, x >__ O,

namely F(x) Ax" for appropriate values of A and n.

Problem 71-2", Brigadients, by ALSTON S. HOUSEHOLDER (University ofTennessee).

Given the two sequences {ai}, {bi}, i- 1, 2,... let /(a)r] denote the deter-L(b)J

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0 a0 at+s_ 2

0 bo br+s-2bo bl br+s-1

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