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Unformatted text preview: ACM116  Winter 20082009  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? 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 , 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 where R k is equal to the payoff factor f if the kth bet is won and is otherwise equal to 0. 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? 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 cases, and station 2 in 2% of cases. Making errors 1 is independent of the actual state of the climatic system. Each station makes its errors independently of the other. At a given time, station 1 is communicating that the system is in the state S 1 whereas station 2 is saying that the system is in state S 2 . Which communication should be assumed to be correct? 5/ I will spin a fair roulette wheel with only five sections. Four of the five sections pay $1; the fifth pays $5. If the cost is $1 . 50 per spin, and you may play as often as you want, should you play the game? 6/ A doctor finds evidence of a serious illness in a particular patient and must make a determination about whether or not to advise the patient to undergo a dangerous operation. If the patient does suffer from the illness in question, there is a 95% probability that he will die if he does not undergo the operation. If he does undergo the operation, he has a 50% probabilitythe operation....
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 Spring '10
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 Normal Distribution, Probability, probability density function, Election Day

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