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

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ACM116 - Winter 2008-2009 - Homework #2 Handed out: 27 Jan 2009, Due: 3 Feb 2009 Please write down your solutions clearly and concisely, put the problems in order, and box your answers. Presentation will be worth a couple of extra points. 1/ If the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ = 3, what is the probability that no accidents occur today? Solution 1 { X = 0 } = e - 3 0 . 05 2/ Suppose for instance that you are offered a sequence of bets, each bet being a losing proposition with probability 1 - p and paying out f times ( f > 1) your stake with probability p . a) Compute the expected net payoff of each bet. (net payoff is simply the amount that you earn minus the amount that you bet) b) Suppose that the expected net payoff of each bet is strictly positive (you have an edge). How to gamble if you must? The idea is to bet a fixed proportion of your present bankroll. When your bankroll decreases you bet less, as it increases you bet more. Assuming that your starting bankroll is V 0 , define the random variable V n as the size of your bankroll after n bets when you bet a fixed fraction α (0 < α < 1) of your current bankroll each time. Here it is supposed that winnings are reinvested and that your bankroll is infinitely divisible. Find the optimal value for α . Hint: Observe that V n = (1 - α + αR 1 ) × · · · × (1 - α + αR n ) V 0 where R k is equal to the payoff factor f if the k -th bet is won and is otherwise equal to 0. 1
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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
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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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