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︰资讯工程系 考试日期（年月日）︰2013/03/25 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:110 You can answer in either Chinese or English 1.[Probability defined on events] Let E and F be mutually exclusiveevents of a random experiment. Suppose that the experiment is repeated until either event E or event F occurs. What does the sample space of this new super experiment look like? What is the probability that event E occurs before event F? (6pts) 2.[set theory and events] If P(A)=0.3, P(B)=0.5 (a) Assume A and B are mutually exclusive, find P(A交集B) and P(A联集B). (4pts) (b) Assume A and B are independent, find P(A'交集b') (5pts) 3.[Probability and Conditional Probability] (a) In a modified Monty Hall Problem, assuming there are 4 doors and behind three of them there are goats, while the remaining one is a car. After a participant picks a door, the host (who knows where the car is) 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 is 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] You have two opponents with whom you alternate play. Whenever you play A, you win with probability pA; whenever you play B, you win with probability pB, where pB > pA. If your objective is to minimize the number of games you need to play before winning two in a row, should you start with A or with B? Prove it. (6pts) 5.[Independency] (a) X, Y, and Z are three random variables. Can you proposal a real-world example of them that satisfy both of the following conditions (5pts): (a) X and Y are independent (2) X and Y becomes dependent given Z (b) Given your grandparent's IQ, is your IQ independent of your father's? (2pts) (c) Given your parent's IQ, is your IQ independent of your sister's IQ? (2pts) 6.[sampling] You are asked to estimate the circumference ratio pi. The only function you can use is the random-value-generator random(). Please write a pseudo code (or java/C) that uses random() to obtain pi. Note that you can use while, if, and +-*/ in the pseudo code. (8pts) 7.[Bayes rule] Assume that there are only Salmons (60%) and Trout (40%) in the sea, and assume the weight and color of a fish are independent. Researches discover the weight (kg) of fishes, W, follows the following p.d.f, f(w) = 5/(11 ((w-mu)^2 +1)) mu = 4 if fish type = "Trout" 5 if fish type = "Salmon" Further, 2/3 of the salmons are black, others are silver; 1/2 trouts are black, the remainders are silver. Today, a fisher catches a black fish with 5kg, what is the probability that this fish is a Salmon? (8pts) 8.[Exponential Distribution] The amount of delay time for a given flight is exponentially distributed with a mean of 0.5 hour, namely, the p.d.f f(x) = 2e^(-2x). Ten passengers on this flight need to take a subsequent connecting flight. 7 persons have connection time for 2 hours, but unfortunately the remaining 3 persons has only 1 hour between flights. (a) Suppose John is one of the 10 passengers needing a connection. What is the probability that he will miss his connection? (5pts) (b) Suppose he met Mike on the plane, who also needs to make a connection. However, Mike is going to another destination and thus has a different connection time from John's. What is the probability that both John and Mike will miss their connections? (5pts) (c) A friend of John's, named Mary, happens to live close to the airport where John makes his connection. She would like to take this opportunity to meet John at the airport. Suppose she has already waited for 30 minutes beyond John's scheduled arrival time. What is the probability that John will miss his connection so that they could have a leisurely dinner together? Assume John's scheduled connection time is 1 hour. (5pts) 9.[Prove Variance] Prove that the variance of an exponential distribution is theta^2 (8pts) 10.[pmf] 1 -lamda lamda^x 1 -mu mu^x A pmf of X is defined as p = － e －－ + － e －－ , 2 x! 2 x! when x=0,1,2..., and p=0 otherwise. (a) Please confirm whether it is a legal pmf (4pts) (b) What is E(X)? (4pts) 11.[mgf] Let R(t) = ln M(t), where M(t) is the mgf of a random variable. Show that (a) R'(0) = mu (4pts) (b) R"(0) = sigma^2 (4pts) 12.[Terminology] Please describe the following terms (a) Theory of Total Probability (5pts) (b) Draw an example pmf so that three random variables X, Y, Z are mutually independent (5pts) (c) What is the relationship between Poisson, Binominal, Chi-Square, Gamma, and Exponential distribution (5pts) ------------------------------------------------------------------------------ Appendix: Poisson Distribution: 略 Exponential Distribution: 略 -- ※ 编辑: danielu0601 来自: 140.112.30.140 (05/06 13:12)