课程名称︰ 机率 课程性质︰ 系必修 课程教师︰ 林守德 开课学院： 电资学院 开课系所︰ 资工系 考试日期（年月日）︰ 97/4/24 考试时限（分钟）： 180 是否需发放奖励金： 是 （如未明确表示，则不予发放） 试题 : (Total Points: 120) 1. Given Poisson arrival rate, if the probability that zero arrival within one hour is 0.1, what is the probability that at least 1 arrives within 30 minutes? (5 points) 2. The joint probability of X and Y looks like: P(x=1, y=2)>0.1, P(x=1, y=3)>0.3, P(x=1, y=1)>0.2, P(x=2, y=1)>0.1, P(x=2, y=2)>0.2, and the rest P(x, y)=0. Is X independent of Y or not? why? (5 points) 3. If A flips an unbiased coin n times and B flips the same coin n+1 times, what is the probability that B has more heads than A? (hint: conditioning on which player has more heads after each has flipped n times.) (7 points) 4. A communication system consists of n components, each of which will, independently, function with probability p. The total system will be able to operate effectively if at least one-half of its components function. Q: find the range of p that makes a 5-component system better than a 3-component system (hint: 2x^3-5x^2+4x-1=(x-1)^2(2x-1)) (5 points) 5. The amount of time (in hours) that a computer functions before breaking down is a continuous r.v. whose p.d.f. is exponential distribution: f(x)=λe^-x/100 when x>=0, and f(x)=0 when x<0. Q1: is the probability that it will function for more than 100 hours larger than 40% or not? (2 points) Q2: if the computer has functioned k hours, what is the probability that it will keep functioning for at least 100 or more hours? (3 points) 6. Your company must make a sealed bid for a construction project. Your company will win if your bid if lower than other companies. If you win the bid, then you plan to pay another firm 100 thousand dollars to do the work. If you believe the minimum bid (in thousands of dollars) of other participating companies can be modeled as a uniform distribution in between (70, 140), then how much should you bid to maximize your expected profit? (10 points) 7. Let X, Y, Z be independent and uniformly distributed over (0, 1) Q1: what is the joint p.d.f. f_xyz(x, y, z) of x, y, z (3 points) Q2: what is the probability P(X>YZ)? (hint: intergrate over f_xyz(x, y, z)) (5 points) 8. The m.g.f. if a r.v. X is (1/4)(e^t+e^2t+e^3t+e^4t) The m.g.f. of a r.v. Y is (1/3)(e^t+e^2t+e^3t), X and Y are independent. Q1: what is the m.g.f of W where W=X+Y? (5 points) Q2: what is P(W=4)? (5 points) 9. Suppose the joint p.d.f. of X and Y is f(x, y)=((e^(-x/y))(e^(-y))/y), 0<x<∞, 0<y<∞ Q1: what is f(x|y)? (5 points) Q2: what is P(X>1|Y=k)? (5 points) 10. Company 1 announces a disease (occur rate: 20%) testing product T1. The performance looks like : P(T1=positive | Disease=true) = 0.7 P(T1=negative | Disease=false) = 0.7 Company 2 also announces a testing product T2 for the same disease. The performance looks like : P(T2=positive | Disease=true) = 0.9 P(T2=negative | Disease=false) = 0.6 Q1: A careless doctor performed a test on a patient and found the result is positive. However this doctor forgot which testing product was chosen. Can you tell this doctor which product is more likely to be the one used given posistive result? (5 points) Q2: If a patient has been tested positive on both products, what is the probablitiy that he/she really has the disease (assuming that the test results are conditionally independent given disease)? (hint: use Baye's rule on P(Disease=true | T1=positive, T2=positive)) (10 points) 11. The p.d.f. of X is f(x)=Θx^(Θ-1), 0<x<1, 0<Θ<∞. Let Y=-2ΘlnX, how is Y distributed? (2 points) The p.d.f. of X is f(x)=Θx^(Θ-1), 0<x<1, 0<Θ<∞. Let Y=x^Θ, how is Y distributed? (hint: F(x)=Y(x)) (3 points) 12. THe test scores of 1000 students show mean=83 and variance=36. At least how many students recieved test scores between 71 and 95 (83 ±12)? (hint: we are asking "at least how many", not "on the average how many") 13. Assuming the IQ of the students in Taipei has mean=110 and variance=225. What is roughly the probability that a randomly chosen student's IQ is larger than 125? (5 points) 14. A random sample X1, X2, ..., Xn is taken from a Poisson distribution with a mean λ. Prove that the maximum likelihood estimator for λ is (X1+...+Xn)/n (5 points) 15. A survey reveals that a candidate will receive 80% of the vote. At least how many people have to return the survey results so we can be 95% confident that the maximum error of this estimation is 1.96% (5 points) 16. The length of 33 sample fish yeild a mean 16.82cm and s^2=34.29. Determine the new sample size so that [16.32, 17.32] is a 95.44% confidence interval for u? (5 points) 17. You have $1024 but need $2048 to buy a game. So you decide to play a blackjack game (assuming winning percentage 0.45) with the hope to win another $1024. Here are two strategies: Strategy 1: Put "all your money in" in one game (if you win, you will bet 2^k after losing k times in a row) Strategy 2: 1. Bet 1 dollars first. 2. If lose, then bet 2. If lose again, then bet 4. You will bet 2^k after losing k times in a row. 3. If win, go back to 1. Stop until you have $2048 in hand or do not have enough money for the next bet. Q: We want to know whether strategy 2 is better than strategy 1 in this case Is the probability of winning $2048 in strategy 2 larger than or smaller than 10^(-1)? (you need to write down how to calculate this probability) (hint: log(0.55)=0.26, log(1-10^(2.6))=-1.09*10^(-3)) (10 points) --

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