课程名称︰工程统计学 课程性质︰必修 课程教师︰朱致远 开课学院：工学院 开课系所︰土木工程学系 考试日期（年月日）︰102/11/15 考试时限（分钟）：110分钟 试题 : Engineering Statistics Midterm Exam (2013/11/5) ● Exam time: 10:20am-12:10pm. ● Closed-book. ● Calculators are allowed. Cell phones are strictly prohibited. ● The calculation process must be described explicitly, including the ● probability functions for the commonly used distributions. ● 30% of the total score. 1. (3%) Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of film at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Jeff, who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages. If a customer complains that her package of film does not show the expiration date, what is the probability that it was inspected by John? 2. (3%) (a) Given a standard normal distribution, find the area under the curve that lies between z=-1.97 and z=0.86. (b) Given a standard normal distribution, find the value of k such that P(k<Z<-0.18)=0.4197. (c) Given a random variable X having a normal distribution with μ=50 and σ=10, find the probability that X assumes a value between 45 and 62. 3. (3%) Show that the mean of a random variable following a geometric distribution is 1/p, where p is the probability of a success. The proof is equivalent to showing that the return period (回归期) of an event occurring with probability p is 1/p. 4. (3%) Let X, Y, and Z be independent variables where E(X)=1, V(X)=3 E(Y)=4, V(Y)=7 E(Z)=3, V(Z)=2 What is the mean and variance of (a)U=3X+4Y (b) V=Y-3Z (c) W=U+V 5. (3%) Flaws occur in mylar material according to a Poisson distribution with a mean of 0.01 flaw per square yard. (a) If 25 square yards are inspected, what is the probability that there are no flaws? (b) What is the probability that a randomly selected square yard has no flaws? (c) Suppose that the material is cut into 10 pieces, each being 1 yard square. What is the probability that 8 or more of the 10 pieces will have no flaws? 6. (3%) Determine the mean and variance of X. 2 f(x)=kx for 0<x<4 7. (6%) Given the joint density function 2 12y , 0<x<1-y,0<y<1 f(x,y)={ 0 elsewhere Determine the following. (a) If X and Y are independent. (b) E(Y|X) (c) P(Y>2|X=0.6) 8. (3%) It is assumed that 4000 of the 10,000 voting residents of a town are against a new sales tax. If 15 eligible voters are selected at random and asked their opinion, what is the probability that at most 7 favor the new tax? (a) Using hypergeometric distribution. (b) Using binomial approximation. 9. (3%) Let X denote the number of bars of service on your cell phone whenever you are at an intersection with the following probabilities: ┌────┬────┬────┬────┬────┬────┬────┐ │x │0 │1 │2 │3 │4 │5 │ ├────┼────┼────┼────┼────┼────┼────┤ │P(X=x) │0.1 │0.15 │0.25 │0.25 │0.15 │0.1 │ └────┴────┴────┴────┴────┴────┴────┘ Determine the following: (a) F(x) (b) Mean and variance (c) P(X<2) --

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