09-10下概率双语B试卷WithAnswertocontinue.doc

10姓名 学号 学院 专业 座位号 ( 密 封 线 内 不 答 题 )密封线线_ _ 诚信应考,考试作弊将带来严重后果！ 华南理工大学期末考试概率统计（双语）试卷B20092010学年度下学期注意事项：1. 考前请将密封线内各项信息填写清楚； 2. 所有答案请直接答在试卷上。 3考试形式：闭卷；满分：100分4考试时间120分钟。题 号123总分分值256015得 分评卷人1FILLING THE BLANKS (25 points)(1) In a bag there are m white balls and n black balls. If we draw one ball from the bag, the probability of the ball being a white one is . If there are three person A,B,C. A draw a ball at first, then B draw a ball, then C draw the ball, if they dont put the ball back into the bag, the probability for C to draw a white ball is m/(m+n) .(2) If the probability density of a random variable X is given by , 3 , the probability for X to take a value between 0.5 and 1.5 is 7/8 .(3) Suppose D(X)=36, D(X-Y)=62, D(X+Y)=42. _-5/24.(4) If a random variable has mean and standard deviation , Chebyshevs theorem tell us that .(5) If we have a normal sample of size n (n is small) with sample mean and sample variance . We will reject the hypothesis with signicant level if .(6) Suppose the pair is independent, have the uniform distribution in the interval . =_.(7) Let be a variable following an exponential distribution with variance , the density function of is .(8) Let , and are independent, is an -distribution with degree and , i.e. .2. COMPUTATIONS: (60 points)(1). There are 4 white balls, 6 black balls in a bag. A ball is drawn at random from the bag and two balls of another color is then put into the bag. Then a second ball is drawn. If the two balls drawn are of the same color, what is the probability that they are both black?Solution: set and as the events of the two balls are all black, white, and of the same color, respectively. Then ,Thus the probability is (2). A supermarket is doing an ad compaign by organizing a lucky raffle draw for every 100 purchasers. The rules is that every one write his own name on a card, put the card into a lottery box, thus there are 100 names in the box. Now every one draw a card from the box without replacement, one by one. The one who draw the card with his own name get a 50% discount on the purchasing. What is the probability that no one will get the discount? (Assume that the 100 purchasers have different names.)Solution: Number the purchaser from 1 to 100. Denote as the events of number get a discount. Then The probability that no one will get the discount is (3). A factory procuces machines. The cost of each machine produced is $ 1000, and the selling price is $ 3000. If the number of X of machines sold has the distributions as follows x0123456P(X=x)0.10.050.10.20.30.150.1How many machines that the factory should produced to maximize the expectation of its profits.Solution: Let be the profit in the case of producing machines, be the sales incomes in the case of producing machines. Then Note that for , but not for . Thus they should produce 4 machines to maximize the expectation of the profits. (4). Let be a ramdom sample of size n from a binomial population , find the estimators of the parameters and by the method of moments. Solution:we have the following formulationsThus are the wanted estimators.(5). A die is so weighted that for any pair of an even number and an odd number, the probability of it shows the odd number is twice of the even number. The rule of the game is that if an odd number is shown, the player lost dollars, if an even number is shown, the player win dollars. On average, how much will the player win for one toss. Solution: Denote the probability to show number as , as the amount the banker win for one toss. we have . Thus And On average, the player loss about $0.66 for one toss.(6). It has been claimed that 50% of the people have exactly two colds per year. We decide to reject this claim if among 400 people 215 or more say that they have had two colds per year. What is the probability that we have committed a type I error?(6 points)Solution: Let be the number of people of this kind. Then approxiamtely we have .And (7) . The mean breaking strength of a kind of cord has been established from considerable experience at 19 ounces with a standard deviation of 1.3 ounces. A factory has a new technology to manufacture this type of cord, and claim that the cord they produced is of the same or greater strength . A sample of 100 pieces obtained from the new machine shows a mean breaking strength of 18.0 ounces. Would you accept the statement at 1% level of significance?Solution: The hypothesis is . If its true, we have . For , But now we have . Thus we reject the hypothesis at 1% level of significance.(8). The time for a kind of glue to set can be treated as a random variable having a normal distribution with mean 29 seconds. Find its variance if the probability is 0.12 that it will take on a value greater than 38.3 seconds.Solution: The time as a variable . Let ,3. PROOFS (15 points)(1). Let be a random sample of size n from a population with mean . Show that is an unbiased estimator of if and only if . And prove that among all estimators of this form, has the smallest variance. Proof: if and only if .Let , we have , and the equality is satisfied if and only if for some . But then we have , thus the discriminant is nonpositive, i.e. . This finishes the proof. (2). Let randon variable be a gamma distribution. To prove that and .(3). In the hypothesis testing problem, if we have random sample of size n from a normal population following , with known.