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# hw2_soln - ACM116 Winter 2008-2009 Homework#2 Handed out 27...

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Solution 2 a) Let s be the amount wagered on a single bet. The expected net payoff is [Net Payoff] = ( f - 1) s (Win) - s ( Lose ) = ( f - 1) sp - s (1 - p ) = s ( fp - 1) (1) Note that for the expected net payoff to be positive, fp - 1 > 0. For example, if p = 1 2 , we must have f > 2, as expected. b) Take the log of V n /V 0 so that we can write our fortune after n trials as a sum of iid random variables: ln V n V 0 = n k =1 ln (1 - α + αR k ) so by the Law of Large Numbers, 1 n ln V n V 0 [ ln (1 - α + αR 1 )] ln ( V n /V 0 ) is an increasing function of V n , so maximizing [ ln (1 - α + αR 1 )] with respect to α maximizes our fortune. Maximizing [ ln (1 - α + αR 1 )] = pln (1 - α + αf ) + (1 - p ) ln (1 - α ) gives α = pf - 1 f - 1 Since pf > 1, f > 1, and p < 1, we have α (0 , 1). Only if p = 1 should we choose α = 1. 3/ You roll two dice at the same time. Each time you get a 6 on a die you should throw it away and roll the other one. Otherwise you keep rolling both of them. The game is over when you throw away both dice. What is the expected number of times you roll? Solution 3 let T n denote the number of times that you roll until you get 6 on all n dice. Conditioning on the event of the first roll and taking expectation from both sides ( T n ) = 1 6 n 5 n ( ( T n ) + 1) + n 1 ! 5 n - 1 ( ( T n - 1 ) + 1) + n 2 ! 5 n - 2 ( ( T n - 2 ) + 1) + · · · + n n - 1 ! 5( ( T 1 ) + 1) + 1 ! 2
for n = 2 this becomes 11 ( T 2 ) = 10 ( T 1 ) + 36 T 1 is a geometric random variable with mean 6. So ( T 2 ) 8 . 73 4/ Two weather stations are giving data on a climate system which can be in two states S 1 and S 2 , shifting at random from one to the other. Long observations have shown that during 30% of the time the system is in the state S 1 and 70% of the time the system in the state S 2 . Station 1 gives erroneous data in 2% of

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