课程名称︰机率 课程性质︰必修 课程教师︰林守德 开课学院：电机资讯学院 开课系所︰资讯工程学系 考试日期（年月日）︰2014/4/14 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 Point : 120 You can answer in either Chiese or English 1.[Independency] X,Y,and Z are three random variables. Can you proposal a real-world example of them that satisfy the following conditions: (a) X and Y are independent, but becomes dependent given Z (5pts) (b) X and Y are dependent, but becomes independent given Z (5pts) 2.[Application on Probability] There is a group of n students who occupied the Legislative Yuan (LY). The observation is that: at each minute, there is 1/2 chance one student will leave LY (i.e. n=n-1), and 1/2 chance one student enters LY (i.e. n=n+1). The police make a deal with the leader: If they reach 2n students before everybody leaves LY (n=0), then the leader will not be arrested. However, if the number reaches 0 before it becomes 2n, then the leader will be arrested immediately. (a) Given t minutes passed, what is the chance that the leader has already been arrested ? (12pts, if you cannot derive the close form solution, please describe how to generate this probability) (b) If t >> n, what is the answer for this question ? (5pts) 3.[Probability and Conditional Probability] (a) In a modified Monty Hall Problem, assuming there are n doors and behind n-k of them are goats, while the remaining k (k << n) is a car. After a participant picks a door, the host (who knows where the cars are) will intentionally open a door with goat. In this case, should the participant swap his current choice with one of the remaining door?(5pts) (b) If the host does not know where are the car, and he opens a door with a goat. Should the participant swap ? (5pts) Please explain your answers using probability. 4.[Probability and expectation] A grocery store has available n watermeleons to sell and makes $1.00 on each sale.Say the number of consumers of these watermelons is a random variable that has a distribution that can be approximated by f(x) = 1/200 , 0 < x < 200, a p.d.f. of the continuous type. If the grocer does not have enough watermelons to sell to all consumers, she figure that she loses $5.00 for each unhappy customer. But if she surplus watermelons, she loses 50 cents on each extra watermelon. What should n be to maximize "profit"? (10pts) 5.[Random Experiment] You are asked to design a random experiment to estimate the circumference ratio π. The only function you can use is the random-value-generator random(), which returns a value between [0,1]. Please describe your experiment (you can use pseudo code or simply explain it in plain text). (10pts, Note that you can use while, if, and +-*/ in the pseudo code) 6.[Bayes rule] Company1 announces a disease (occur rate = 20%) testing product T1. The performance looks like: P(T1 = positive | Diease = true ) = 0.7 P(T1 = negative | Diease = false) = 0.7 Company2 also announces a testing product T2 for the same disease. The performance looks like: P(T2 = positive | Diease = true ) = 0.9 P(T2 = negative | Diease = false) = 0.6 Q1: A careless doctor performes a test on a patient and found that the result is positive. However,this doctor forget which testing product was chosen. Can you tell this doctor which product is more likely to be the one used given positive result ? (5pts) Q2: If a patient has been tested positive on both products, what is the probability that he/she really has the diease (assuming that the test results are conditionally independent given diease) ? (5pts) 7.[Exponential Distribution] Let X be a random variable that represents the number of days that it takes a high-risk driver to have an accident. Assume that X has an exponential distribution. If P(X<50)=0.25, compute P(X>100 | X>50). (5pts) 8.[Poisson] Given the following random experiments, please comment whether each of them is likely to produce a random variable that follows a Poisson distribution, and explain why : (a) Observing the number of people entering CSIE R104 front door from 14:20-15:00 every Monday (b) Observing the number of cars passing 长兴街警卫亭 every Monday from 10-11am (c) Observing the number of cars passing 新生南路忠孝东路交叉口 at 5-6pm every Monday (9pts) 9. Amy wants to buy the May-Day concern ticket online. Since it is really hard to get one, she used five computers (C1~C5) to in pareallel to ensure she can get it. But there are different successful rate for each computer to get the ticket. The successful rate of Ci computer is P(i+1) = 1.05*P(i)-0.05. A 'round' is defined as one trail for every computer. A random variable X is defined as the number of rounds required to get one ticket. We know the variance of X is 2. So what's P(1) ? (7pts) 10. In the movie theater there are 300 seats. 300 people including Jack lined up to enter the theater. Jack is the 200th person to enter the theater, but when it's his term, he found that his seat number faded so he has no idea where to seat. Jack decides to randomly pick the seat because everyone was so nice that if their seat is taken, they will simly find other seat randomly. What's the probability of the last person to seat his/her own seat correctly ? (10pts) 11. One day Takasi found that around him there were four zombies. He picked up the gun to kill them. Since it becomes easier to kill a zombie if there are more around, the probability for Takasi to kill the zombie if n/5 (n is the number of zombie) and he can only kill one zombie in a time. When the zombie was killed, the zombie will have 30% chance to revived by their friend. That says, if there is only one zombie left, it won't rebirth. What's the expected time Takasi needed to kill all the zombies ? (10pts) 12. In the MMORPG world, Shiroe and his partners wants to meet with the king. They started at "A" and is allowed to choose only three directions (right, up, down), with probability becomes P(right) = 1/2, P(up) = P(down) = 1/4. However, if he can't go up or down, the probability becomes P(right) = 1/2, P(up or down) = 1/2. In the graph there are many traps in each area, denoted as 'X'. Entering 'X' means game over. One special place is the rest area, there they can choose to start at "B", "C" and "D". Assuming Shiroe took SD's probability couse previously and can make the best decision, what's the probability that Shiroe meets the king safely ? (12pts) ┌─┬─┬─┬───┬─┬─┬─┬───┐ │O │O │O │ │B │O │O │ │ ├─┼─┼─┤ ├─┼─┼─┤ │ │A │O │X │ Rest │C │O │X │ King │ ├─┼─┼─┤ ├─┼─┼─┤ │ │O │O │O │ │D │O │O │ │ └─┴─┴─┴───┴─┴─┴─┴───┘ ------------------------------------------------------------------------------ Poisson distribution λ^x e^-λ f(x) = ───── x ! μ = λ, σ^2 = λ (1-p) The variance of geometry distribution is f(x) = ──── p^2 Exponential Distribution f(x) = λe^(-λx), let θ = 1/λ, μ = θ , σ^2 = θ^2 --

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