課程名稱︰機率 課程性質︰必修 課程教師︰林守德 開課學院：電機資訊學院 開課系所︰資訊工程系 考試日期（年月日）︰2013/05/06 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 : Probability 2013 Midterm2 (Prof. Shou-de Lin) 5/6/13 14:30-17:30pm Total Points:110 You can answer in either Chinese or English 1. Short answer: (a) provide the definition for the following term "law of large number" (b) how large normally do we need for n to make the central limit theory work? (6pts) x -λ 2. A random variable X follows Poisson distribution: Poisson(λ) = (λ) e /x! and given X = x, a random variable Y follows binomial distribution: xCy p^y (1-p)^(x-y) Hence, given Y = y, show that the conditional distribution P(X|Y=y) is also Poisson distribution (8pts) 3. (a) We are given a function random() that generates a random variable X following gamma distribution with mean = 2 and variance = 2.25. Using random(), please write the pseudo code to produce some values of Y that approximately follows N(mean = m, variance = v). (5pts) (b) Write a pseudo code to produce the same Y in (a), but this time assume the random() function only returns a uniformly distributed random variable between 0 and 1. (5pts) 4. The probabilities of earthquakes occurrence in Taipei and Hualien are independent. The Poisson random variables X (parameter = λ1) and Y (parameter = λ2) represent the total number of earthquake appearance a certain interval in Taipei and Hualien respectively. If λ1 = 3/hour and λ2 = 5/hour and we have observed totally 4 earthquakes in an hour, what is the probability that three of them come from Hualien? (8pts) 5. Suppose that X1, X2 are independent Poisson random variables with respective parameters λ1, λ2. Find Cov(X1 + X2, X1 - X2), where Cov(X1,X2) = E[(X1-μ1)(X2-μ2)] (8pts) 6. Let {X1, X2, ...} be a sequence of independent Poisson random variables, each with parameter 1. By applying the central limit theorem to this sequence, prove that 1 n n^k 1 (8pts) lim －－－ Σ －－ = － n->∞ exp(n) k=0 k! 2 7. A point (X, Y) is selected randomly from the triangle with vertices (0, 0), (0, 1) and (1, 0). Calculate E(X|Y=y) (8pts) 8. Let X and Y be two independent random points from the interval (0, 1). Calculate the CDF of max(X, Y)/min(X, Y) (10pts) 9. X is a discrete random variable with pmf f(x) = exp(-λx)-exp(-λx), where X∈[0, ∞] (1) Show that f(x) is a legal pmf function (3pts) (2) Is X a memoryless random variable? Why? (6pts) 10.(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) 11.X and Y are two discrete random variables. We sample them 100 times total and found the outcomes look like below. What is the correlation coefficient between X and Y (Note:ρ=Cov(X,Y)/σ1σ2): (7pts) outcome X = 140 X = 210 X = 280 Y = 90 10 times 10times 10times Y = 180 10 times 20times 10times Y = 270 10 times 10times 10times 12.Let {X1, X2, X3,...,Xn} be a sequence of i.i.d. exponential random variables with parameter θ. Find the distribution function of Σi=1~n Xi. (10pts) 13.Your company must make a sealed bid for a construction project. Your company will win if your bid is lower than other companies. If you win the bid, then you plan to pay another firm 100 thousand dollars to do the work. If you believe the minimum bid (in thousands of dollars) of other participating companies can be modeled as a uniform distribution in between (70,140), then how much should you bid to maximize your expected profit? (10pts) --

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