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Unformatted text preview: EE376B/Stat 376B Handout #17 Information Theory Tuesday, May 31, 2011 Prof. T. Cover Solutions to Homework Set #7 1. Growth rate. Let X = { (1 ,a ) , with probability 1 / 2 (1 , 1 /a ) , with probability 1 / 2 , where a > 1. This vector X represents a stock market vector of cash vs. a hot stock. Let W ( b ,F ) = E log b t X , and W * = max b W ( b ,F ) be the growth rate. (a) Find the log optimal portfolio b * . (b) Find the growth rate W * . (c) Find the asymptotic behavior of S n = n i =1 b t X i for all b . Solution: Doubling Rate. (a) Let the portfolio be (1 b 2 ,b 2 ). Then W ( b ,F ) = 1 2 ln(1 b 2 + ab 2 ) + 1 2 ln(1 b 2 + b 2 a ) . (1) Differentiating to find the maximum, we have dW db 2 = 1 2 a 1 1 b 2 + ab 2 1 1 a 1 b 2 + b 2 a = 0 (2) Solving this equation, we get b * 2 = 1 2 . Hence the log optimal portfolio b * is ( 1 2 , 1 2 ). 1 (b) The optimal doubling rate W * = W ( b * ,F ) is W * = 1 2 ln ( 1 2 + a 2 ) + 1 2 ln ( 1 2 + 1 2 a ) (3) = 1 2 ln (1 + a ) 2 4 a .....
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