課程名稱︰統計學上 課程性質︰必修 課程教師︰關秉宗 開課學院：生農學院 開課系所︰森林環資系 考試日期（年月日）︰2011/11/10 考試時限（分鐘）：180 mins 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Fall 2011 Statistics Midterm Examine November 10, 2011 1. Given three events A1, A2, and A3 such that P(A1) = 0.3, P(A2) = 0.6, P(A3) = 0.4; P(A1∩A2) = 0.2, P(A1∩A3) = 0.1, P(A2∩A3) = 0.3; and P(A1∩A2∩A3) = 0.05. Please find (a) P(A1∪A2∪A3) and (b) P(A1'∩A2'∩A3') (10%) 2. A boy has 5 blue and 4 white marbles in his left pocket and 4 blue and 5 white marbles in his right pocket. He randomly transfers one marble from his left pocket to his right pocket. He then draws a marble randomly from his right pocket. What is the probability that the marble he drew will be a blue one? (10%) 3. Life tables show the average numbers of survivors at various ages per 100,000 live births. Age 20 45 65 Males 97108 92191 71385 Females 98040 95662 84483 If we assume that mortality rates are constant through time, we can use these numbers to estimate survival probabilities. For example, the probability of that a newborn boy live to age 65 is 0.71385, and a male aged 45 lives to age 65 is about 71385/92191. Please use these data to find the probabilities of (a) a woman aged 20 will live to age 65 (5%) (b) a man and a woman aged 20 will both live to age 65 (5%) (c) a person aged 45 is male (10%) (d) a person aged 45 will live to age 65 (15%) Note: Assume that 52% of the live births are males and 48% are females. 4. A die is rolled twice. Let the probability of the each faces be 1/6 for each roll. Let A be the event that the first roll shows a number <= 2(小於等於2), and B be the event that the second roll shows a number at least 5. Please find the probabilities of A, B, A∩B, A∪B, and P(B|A). Are A and B independent? (10%) 5. Please find the mean and variance of the following p.d.f. (10%) (a) f(x) = 1/5, x = 5, 10, 15, 20, 25; (b) f(x) = 1, x = 5; 3! 1 x 3 3-x (c) f(x) = ────(─) (─) ,x = 0, 1, 2, 3 x!(3-x)! 4 4 (註：後兩項乘項為四分之一的x次方乘上四分之三的(3-x)次方) 6. Let X have a Poisson distribution with a mean of 4, Please find (10%) (a) P(2 <= X <= 5) (b) P(X <= 3) (c) P(X >= 3) 7. Please prove that the sum of binomial pdf is 1. (15%) Please show the details. --

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