课程名称︰机率 课程性质︰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）︰2012/5/14 考试时限（分钟）：180分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Probability 2012 Midterm2 (Prof. Shou-de Lin) 5/14/12 14:30-17:30pm Total Points: 120 You can answer in either Chinese or English. 1. A and B are events. For any A and B, can you generate the conditional probability P(X A | Y B) using the conditional probability distribution f(x|y)? Explain your answer. (7 pts) 2. You have entered the 'ultimate gamble' with X dollars, the rule is as followings: ● In each round you have to bet all your money. If you win, your money will be doubled. If you lose, you money will become half. If the total amount of money reaches X/4, you will be kicked out of the game and cannot play anymore. Please write a pseudo code that outputs the following probability value: ● Assuming the winning percentage is p, what is the probability that you will be kicked out before your money become n*X? (7 pts) 3. If X1, ... , Xn are identically independent distributed Poisson(λ), define Y = X1 + ... + Xn, what is E[X1 | Y=y]? (6 pts) 4. Let X~N(μ,σ^2), a lognormal distribution is Y:log(Y)~X. What is the mean and variance of Y? (8 pts) 5. Let X have a logistic distribution with pdf f(x) = exp(-x)/(1+exp(-x))^2, x can be any real number. What is the distribution of Y = 1/(1+exp(-X)) (7 pts) 6. Y follows Bernoulli distribution where p=0.5, and X follows a normal distribution with mean equals y and variance 1. What is the pdf of X? Please draw it. (6 pts) 7. There are two kinds of bulbs in the world, type A and B. Let A-bulb and B-bulb represent type A bulb and type B bulb respectively. The life time of A-bulb is N(2,16) and B-bulb is N(4,16). John designs a device that will automatically turn on the next bulb when the previous bulb burnt out. That is, it will turn on the 2nd bulb when 1st bulb burnt out, turn on the 3rd bulb when 2nd bulb burnt out, and so on. Now, he put x A-bulbs and y B-bulbs on that device. After doing that, he found 95% confidence interval of the lighten time of that device is 88 ± 203.84. What are x and y? (Z(2.5%)=1.96) (10 pts) 8. In a modified Monty Hall Problem, assuming there are 5 doors and behind 4 of them there is a goat, while the remaining one is a car. The participant is allowed to pick two doors, and he/she will win the car if any of the door indicates a car. (12 pts) ● After CCF picks two door, the host (who knows where the car is) intentionally opens another door (among the remaining doors) with a goat. CCF is given a choice of swap his two doors with the remaining unopened doors. Unfortunately, he is too dumb to know what to do, can you tell him if he should swap? Explain your answer. ● After CCF picks two door, the host (who knows where the car is) intentionally opens one of his chosen door with a goat. He then offered two choices as below. Which should he choose? Justify your answer. 1. keep the other chosen door, and pick one from the remaining three doors. 2. don't keep the other chosen door, and pick two from the remaining three doors. 9. The owner of a property that is for sale is willing to accept the maximum of four independent bids (in $100,000 units), which have a common p.d.f. f(x) = 2x, 0 < x < 1. What is the expected value of the highest bid? (10 pts) 10. Let Xi be a distribution with mean u and variance d^2, Y = 1/n*(X1+X2+...+Xn). Prove that P(|Y-u| ≧ d) ≦ 1/(n^2) (8 pts) 11. Companyl announces a disease (occur rate= 20%) testing product T1. The performance looks like: (9 pts) ● P(T1=positive | Disease=true) = 0.7 ● P(T1=negative | Disease=false) = 0.7 Company2 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 that 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 positive result? (5 pts) Q2: If a patient has been tested positive on both products, what is the probability that he/she really has the disease (assuming that the test results are conditionally independent given disease)? (5 pts) 12. CCF repeatedly roll a fair die. If it comes up 6, he instantly win (and stop playing); if it comes up k, for any k between 1 and 5, he waits for k minutes and then roll again. What is the expected elapsed time from when he start rolling until he wins? (Note: If I win on my 1st roll, the elapsed time is zero.) (7 pts) 13. n follows Poisson with mean 25. What is the upper bound and lower bound of p(n>40)? (5 pts) 14. Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0,12). Given X = x, Y is uniformly distributed on the interval (0,x). Determine Cov(X,Y) according to this model. (8 pts) 15. Let X, Y, Z be independent and uniformly distributed over (0,1) ● Q1: what is the joint pdf of x,y,z f(x,y,z)? (3 points) ● Q2: what is the probability P(X>YZ)? (6 points) Appendix: Poisson Distribution λ^x * e^(-λ) f(x) = ─────── , μ = λ, σ^2 = λ, Mx(t) = exp(λ(e^t - 1)) x! Normal Distribution 1 -(1/2)[(x-μ)/σ]^2 f(x;μ,σ^2) = ───── * e σ√(2π) Mx(t) = exp(μt + (1/2)*(σ^2)(t^2)) --

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