課程名稱︰機率 課程性質︰必修 課程教師︰林守德 開課學院：電資院 開課系所︰資訊系 考試日期（年月日）︰2011/05/12 考試時限（分鐘）：180 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Discribe central limit theory (5pts) 2. X, Y, and Z are three random variables. Can you propose a real-world example of them that satisfies both of the following conditions (6pts): ． X and Y are independent ． X and Y becomes dependent given Z 3. Prove Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y) (8pts) 4. (Textbook 5.6-14) Let X and Y equal the numbers of hours a ramdomly selected child watches movies or cartoons on TV during a certain month. Assuming E[X]=30, E[Y]=50, Var(X)=52, Var(Y)=64, and Cov(X,Y)=14. 25 children are selected at random. Let Z equal the total number of hours these 25 children watch TV (either movies or cartoons) in the next month, approximate P(1970<Z<2090) (8pts) 5. (Textbook 5.3-8) Suppose two independent claims are made on two insured homes, where each claim has p.d.f. f(x) = 4/x^5, 1 < x < ∞ Find the expected value of the larger claim. (8pts) 6. Let X1, X2 be two independent R.V. with Poisson distribution with parameters λ1, λ2, find the distribution of Z = X1 + X2 (8pts) 7. Let X1, X2,..., Xn be iid with normal density N(μ,σ^2). Find the distribution of: n n Σ kXk - μΣ k k=1 k=1 Yn = ───────── (10pts) n Σ k^2 k=1 8. Two students A and B one day were arguing whether Prof. CYY is more handsome than Prof. HTL or not. Of course, it is very difficult to reach a conclusion so they decide to resort to voting. This time they want to use "contrastive survey" (對比式民調). This survey asks 100 persons this question "Whether you believe Prof. CYY is more handsome than Prof. CCF" and another 200 person this question "Whether you believe Prof. HTL is more handsome than Prof. CCF". It turns out that CYY:CCF=75:25 while HTL:CCF=70:30. To claims that CYY's supporting rate is 5% (0.75-0.7 = 0.05) higher than HTL, can you calculate the 90% confidence interval for this claim? (12 pts) 9. You are asked to design a random experiment to estimate the circumference ratio π. The only function you can use is the random-value-generator random(), which returns a value between [0,1]. Please describe your experiment (you can use pseudo code or simply explain it in plain text). Hint: we have talked about such experiment last week during the class. (10 pts) 10.Using only random(), which returns a value between [0,1], to generate a sequence of values that follows pdf f(x)=e^-x/(1+e^-x)^2, please write a psuedo code to do that (8 pts) 11.In front of you there are 3 doors and you have to create a method to choose them randomly but equally. You are given only a biased coin whose P(H)≠P(T). Can youuse this coin to design a random experiment whose outcome maps equally distributed decisions? (8pts) 12.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). Determine Cov(X,Y) according to this model (10 pts) 13.A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease present and 0 for disease not present. Let X denote the disease state of a patient, and let Y denote the outcome of the diagnostic test. The joint probability function of X and Y is given by: --------------- | X | |---------------| | 0 | 1 | -----+-------+-------| | |0| 0.800 | 0.05 | | Y |-+-------+-------| | |1| 0.025 | 0.125 | --------------------- Calculate the conditional variance of the outcome of the diagnostic test, given that the patient has the disease. (8 pts) 14.Prof. Lee spent her honeymoon in Japan. The daytime temperature of this country fluctuates every day with sample mean 85 F. (a) Prof. Lee says she is 95% confident that during her stay the true mean temperature was between 84F and 86 F (assuming the true variance is 4). Then how long should this vacation be for her statement to be ture? (8pts) (b) Assuming 85F is the true mean and 4 is the true variance. Prof Lee says she is "at least" 95% certain that the average temperature during her stay was between 84F and 86 F. Then how long should this vacation be for her statement to be true? (8 pts) --

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