課程名稱︰機率 課程性質︰必修 課程教師︰林守德 開課學院：電資 開課系所︰資工 考試日期（年月日）︰3/31 考試時限（分鐘）： 180分鐘 是否需發放獎勵金： （如未明確表示，則不予發放） 試題 : 1.There are three TAs in the probabilityclass. Assume all the midterm exam papers are graded by the same TA, and each Ta has 1/3 chance to grade them. Also each TA has the same 95% chance to grade a paper fairly. TA1 has 1% chance to giv a higher score than the actual score, while TA2 has 2% and TA3 has 3% chance to do the same thing. Given every paper is graded fairly, what is the probability the exam is graded by TA1?(10 pts) 2.(a)In a modified Monty Hall Problem, assuming ther are 4 doors and behind three of them there is a goat, while the remaining one is a car. When a participant SD picks a door, the host(who knows where the car is) intentinally opens a door with goat. SD is given a choice of swap, should he?(7pts) (b) If the host does not know where is the car, and he opens a door with a goat. Should SD swap? (7pts) 3.The owner of a property that is for sale is willing to accept the maximum of four independent bids (in $1000,000 units) , which hav a common p.d.f. f(x) = 2*x, 0<x<1.What is the expected value of the highest bid?(15pts) 4.Toss two different standard dice, white dice and black dice, and let A1={first die=1,2 or 3},A2={first die =3,4 or 5},A3={sum of faces is 9}. (a)What is P(A1),P(A2),P(A3)? (2pts) (b)What is P(A1A2A3)? (2pts) (c)What is P(A1A2)? (2pts) 5.Does mutual independence (i.e. P(A1A2A3)=P(A1)P(A2)P(A3)) imply pairwise independence? If yes, prove it. If not, propose a counter example. (10pts) 6.If E[X^r] = 5^r,r=1,2,3..., find the moment-generating function M(t) of X and the p.m.f of X.(12pts) 7.A store has ordered five copies of a certain issues of a photography magazine. If X has a Poisson distribution with parameter λ=4 , what is the expected number of copies that are sold (note: you don't need to generate the final value, just list the equation is good enough) (8pts) 8.Suppose a department offers three probability classes: basic probability has 80 registered students, advanced probability has 15, and special topic in probability has 5 students. The department chair than announced that the average class size is (80+15+5)/3=33.3. However,Prof. SD argues that this is an underestimate of the class size. The true average class size should be roughly doubled. Can you explain why? (10pts) 9.A grocery store has available n watermelons 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 figures that she loses $5.00 for each unhappy customer. But if she has surplus watermelons, she loses 50 cents on each extra watermelon. What should n be to maximize "profit" ? (11pts) 10.A wireless sensor that contunuously measures and records radiance is placed in a remote region. The time, T,to failure of this sensor is exponentially distributed with mean 3 years. Since the sensor will not be monitored during its first two years of service, the time to discovery of its failure is X= max(T,2). Determine the expected time to discover the sensor's failure? (12pts) 11.Please describe how to estimate the value of e (i.e 2.71828...) using only a random function r() (i.e. returns a real number between [0,1]) with +,-,*,/. Please write a C or pseudo code to do so (hint: there is 'e' in Poisson distribution) (12pts) 註:formula given Poisson Distribution Exponential Distribution --

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