1.Suppose that X and Yare two continuous random variables with the joint probability density function /(x.y) - x + 2y for 0 ＜x＜2 and 0 ＜y ＜ 2. The covariance of Xand Y is________.

2.A random variable X has a mean p = 5 and variance σ2 = 36. Based on Chebyshev's inequality, a lower bound on Pr(-7＜X＜ 17)is____________.

3.Suppose that X is a father's height, Y is a mother's height, Z is a child's ultimate height (as an adult), and G is delined as: G = I if the child is a boy and G = -I ifthe child is a girl. Let M = (X+ Y)/2. Suppose that, conditional on known values of G, X, and Y, Z has a continuous uniform distribution with range [M -5 +3G, M + 5 + 3G]. If a girl's father is 4 inches taller than her mother, the probability that she will grow to be taller than her mother is__________.

4.A manufacturer of car tires claims that his tires will last, on the average, 4 years with a standard deviation of 1 year. Given that 5 of these tires have lifetimes of 2.1, 2.2, 3.0, 3.6, 4.1 years, if someone wants to test whether his tires have a standard deviation of1 year, the figure of the appropriate test statistic is equal to__________.

5.You are going to carry out an experiment to determine whether surface finish has an effect on the endurance limit of steel. It is expected that polishing increases the average endurance limit. In such an experiment, you want to detect that polishing fails to have an effect with a probability of 0.99 (i.(E), a = 0.01) and you also want to detect a change in the average endurance limit of 700 units by a probability of at least 0.9. If it is known that the standard deviation of the endurance limit of the steel is 420 units, then the sample size that would be needed to perform this experiment is____________ . Assume that for a = 0.01, Za = 2.5 and forβ -0.10, 2β- 1.5.

6.How large a sample must be taken in order that you are 95 percent certain that is within 0.5 of the standard deviation of the mean?____________.

7. Ten trials are conducted of a given system with the following results: S, S, F, S, S, S, F, S, S, S ("S" represents success and "IF" represents failure), assuming that the trials are independent of each other. The maximum likelihood estimate of the probability of successful operation is___________.

8.At the beginning of the schoo! year, the mean score of a group of 49 sixth-grade students on an achievement test in reading was 70 with an unbiased sampie variance of 36. At the end of the school year, the mean score on an equivalent form of the same test was 74 with an unbiased sample variance of 25. The correlation between scores made on the initial and final testing was 0.6. To test whether the mean scores on these two exams are significantly different based on the Student t-test, the figure of the test statistic is equal to___________.

(4)之機率為何?

(2) 請問之分配為何?

七、為了估算台北地區家戶持有車輛的狀況,有兩組研究人員分別進行一項隨機抽樣調查,其中一組研究人員估計台北地區家戶未擁有任何車輛之比例約 為62%,另一個則估計約為42%,請問上述兩組研究人員之推估值,在95% 信心水準下,欲達估計誤差值不大於12%之精度,所需之樣本量各應為多少? 您認為此一研究所需之樣本量應為多少較佳?理由何在? 

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