課程名稱︰機率導論 課程性質︰必修 課程教師︰陳 宏 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2013/05/09 考試時限（分鐘）：13:20PM-15:10 試題 : Introductory Probability 機率導論 期中考特考 考試時間: 13:20PM-15:10 Thursday, May 9th, 2013 共215分，最高得分為180分。解答過程須詳列。 當題目中所提供的參考答案，都與你的答案不同時，請註明以上皆非。 1. (points) Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0,12). Given X = x, Y is uniformly distributed on the interval (0,x). (a) (15 points) Derive joint density function of (X,Y). (b) (15 points) Determine Cov(X,Y) according to this model. 參考答案為(A)0(B)4 (C)6(D)12(E)24。 2. (20 points) A device contains 兩個線路板. 第二個線路板 is a backup for the first, 所以只當第一個線路板異常時，第二個線路板才會啟動。 故the device fails 僅當第二個線路板也異常。 令 X 及 Y 分別為第一個線路板的異常時間及第二個線路 板的異常時間， X and Y have joint probability density function ╭ │6e^-x ˙ e^-2y for 0 < x < y < ∞ f(x,y) = ﹤ │0 otherwise ╰ What is the expected time at which the device fails? 參考答案為(A)0.33(B)0.50(C)0.67(D)0.83(E)1.50。 3. (20 points) A device that continuously measures and and records 地震活動 is placed in a remote region. The time, T, to failure of this device is exponentially distributed with mean 3 years. Since the device will not be monitored during its first two years of service, the time to discovery of its failure is X = max(T,2). Determine E[X]. 參考答案為(A) 2+(1/3)exp(-6) (B) 2-2exp(-2/3 + 5exp(-4/3) (C)3 (D)2 + 3exp(-2/3) (E)5。 4. (30 points) A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extened warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold, and suppose that ╭1/6 for (x,y) = (0,0), │1/12 for (x,y) = (1,0), │1/6 for (x,y) = (1,1), P(X = x, Y = y) = ﹤1/12 for (x,y) = (2,0), │1/3 for (x,y) = (2,1), │1/6 for (x,y) = (2,2). ╰ What is the variance of X? 參考答案為 (A)0.47(B)0.58(C)0.83(D)1.42(E)2.58 5. (20 points) A device contains two components. The device fails if either component fails. The joint density function of the life times of the components, measured in hours, is f(s,t), where 0<s<1 and 0<t<1. What is the probability that the device fails during the first half hour of operation? 寫出其數學表達式。 6. (30 points) The stock prices of two companies at the end of any given year are modeled with random variables X and Y that follow a distribution with joint density function ╭ │ 2x for 0 < x < 1, x < y < x+1, f(x,y) = ﹤ │ 0 otherwise. ╰ What is the conditional variance of Y given that X = x? (A)1/12(B)7/6(C)x + 1/2(D)x^2 - 1/6(E)x^2 + x + 1/3 7. (30 points) A company prices its hurricane insurance, 颶風保險, using the following assumptions: (i) In any calendar year, there can be at most one hurricane. (ii) In any calendar year, the probability of a hurricane is 0.05. (iii) The number of hurricanes in any calendar year is independent of the number of hurricanes in any other calendar year. 在該公司的假設條件之下， (a) (15 points) Calculate the probability that there are fewer than 3 hurricanes in a 20-year period. (b) (15 points) Give an approximation of (a) in terms of a Poisson random variable with mean λ. 8. (15 points) Let C_1, C_2, and C_3 be independent events with probability c c 1/2, 1/3, 1/4, respectively. Compute P(C_1∩C_2|C_3∪C_2). 9. (20 points) Let X be a Poisson random variable with parameter λ. Find a simple expression of E[1/(1+X)]. --

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