課程名稱︰機率 課程性質︰系必修 課程教師︰林守德 開課學院：電資學院 開課系所︰資訊系 考試日期（年月日）︰2010/05/13 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. A system contains a component and a backup unit, and each has mean lifetime μ. If the component fails, the sustem automatically substitutes the backup unit, butt there is some chance p < 1 that such substitution fails. Please find the expected lifetime for this system. (7 points). 2. What is the main difference between a random variable and a variable in a programming language (e.g. C)? (5 points) 3. X is a discrete random variable with pmf f(x) = exp[-λx] - exp[-λ(x+1)], where X belongs to [0, ∞] (1) Show that f(x) is a legal pmf function (3 pts) (2) Is X a memoryless random variable? Why? (7 pts) 4. Prove Ex[Ey[Y|X]] = E[Y] (10 pts) 5. X and Y are two discrete random variables. We sample them 100 times total and the found the outcomes look like: outcome X = 140 X = 210 X = 280 Y = 90 10 times 10 times 10 times Y = 180 10 times 20 times 10 times Y = 270 10 times 10 times 10 times What is the correlation coefficient between X and Y? (10 pts) (Note: ρ = Cov(X, Y)/(σx*σy), Cov(X, Y) = E[X, Y] - μx * μy Hint: you don't really have to do complex calculation.) 6. The owner of a property that is for sale is willing to accept the maximum of four independent bids (in $100,000 units), which have a common p.d.f. f(x) = 2x, 0 < x < 1. What is the expected value of the highest bid? (8 pts) 7. Let X have a uniform distribution on the interval (0,2). Given X = x, let Y have a uniform distribution on the interval (0,x). (a) Define the conditional p.d.f. of Y, given that X = x. Be sure to include the domain. (5 pts) (b) Find E(Y|x). (5 pts) 8. A fair coin is tossed until it comes up head for the 20th time. Use the central limit theorem to estimate the probability that more than 50 tosses are needed. (10 points) 9. Prof. HT spent his honeymoon is Czech Republic. The daytime temperature of this country fluctuates every day with sample mean 85F. (a) Prof. HT says he is 95% confident that during his stay the ture mean temperature was between 84F and 86F (assuming the ture variance is 16). Then how long should this vacation be for his statement to be ture? (7 pts) (b) Assuming 85F is the ture mean and 16 is the ture variance. Prof. HT says he is "at least" 95% certain that the average temperature during his stay was between 84F and 86F? Then how long should this vacation be for his statement to be ture? (8 pts) 10.Suppose we are calculating the volume of an n-dimensional box (the volume is just the product of its side lengths). Let Xi be the i-th side length, and Xi's are iid, uniformly distributed in the interval [0,1]. Please find the pdf of the box's volume for sufficiently large n. (15 points) 11.Warning, warning!! The probability aliens attack Star Trek spaceship again!! Similar to last time, they disable all functions of your spaceship except the random function (this function returns a random value between [0,1]). They ask you, the chief scientist of the Star Trek spaceship, to accomplish the following two tasks before 5/13/2010 17:20pm, or the spaceship will explore (along with your midterm score). (a) generate a sequence of values that follows pdf f(x) = exp[-x]/(1+exp[-x])^2 (Note: you are allowed to use logarithm function in this sub-problem, 10 points) (b) estimate the value of e (10 points) Please describe in plain text or in C language how to achieve these tasks. Useful formulas: Negative Binomial Distribution pmf g(x) = (x-1)C(r-1) * p^r * (1-p)^(x-r), x = r, r+1, ... Geometric Distribution pmf g(x) = p*(1-p)^(x-1), x = 1, 2, 3, ... mean = 1/p, variance = (1-p)/p^2 Integration ∫ln(x) dx ∫(ln(x))^2 dx Standard Statistical Tables Areas under the Normal Distribution --

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