[試題] 100-2 呂學一機率第一次期中考消失

看板NTU-Exam作者harry2145時間12年前 (2012/03/26 19:34)推噓3(3推 0噓 1→)

留言4則, 4人參與討論串1/1

課程名稱︰機率課程性質︰必帶課程教師︰呂學一開課學院：電資學院開課系所︰資訊系考試日期（年月日）︰2012/03/26 考試時限（分鐘）：180 是否需發放獎勵金：是（如未明確表示，則不予發放）試題 : 說明：總共十二題，每題十分，可按任何順序答題。歡迎使用事先準備的A4大抄。每題難度不同，請審慎判斷恰當的解題順序。第一題 Sixty percent of the families in a certain community owns their own car , thirty percent own their own home, and twenty percent own both their own car and their own home. If a family is chosen uniformly ar random from the community, what is the probability that this family owns a car or a house but not both？第二題 Suppose that we have ten coins which are such that if the i-th coin is flipped then heads will appear with probability i/10, i = 1,2,...,10. When one of the coins is selected uniformly at random and flipped, it shows head. What is the probability that it was the fifth coin？第三題 Let F be an event with P(F)>0. Define function Qf by Qf(E)=P(E|F) for any enent E. Prove that Qf is a probability function. 第四題 Let X ba a nonnegative integervalued random variable. Prove that ∞ E[x] = Σ P(X≧i). i=1 第五題球桶裡有一顆紅球一顆黑球。每一回合我們等機率從桶中摸出一顆球並立刻放回去，然後多加一顆跟剛剛那顆同色的球。例如第一回合如果摸到紅球，在第一回合結束的時候，就會有兩顆紅球和一顆黑球在桶子裡。請問在第n回合結束的時候，桶子剛好有i顆紅球的機率是多少？其中i可以是1,2,...,n+1中的任何一個整數。也請證明你的答案是正確的。第六題 If E[x] = 2 and Var(X) = 7, compute E[(4-x)^2] and Var(4-3x). 第七題 An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up？第八題 Let X be a Poisson random variable with parameter λ. Show that P(X=i) increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer not exceeding λ.(Hint： Consider P(X=i)/P(X=i-1).) 第九題 Suppose that P(X=a) = p and P(X=b) = 1-p. Please prove that X-b/a-b is a Bernoulli random variable. You are also asked to compute Var(X). 第十題 Independent trials consisting of rolling a pair of fair dice are performed. What is the probability that an outcome of 6 appears before an outcome of 8 when the outcome of a roll is the sum of the dice？第十一題 If X is a geometric random variable, prove that P(X=n+k | X>n) = P(x=k). 第十二題請說明為甚麼Poisson distribution 可以看成是 binomial distribution 的一種極限。 -- ※ 發信站: 批踢踢實業坊(ptt.cc)

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03/27 13:44, , 1^F

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andy74139

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