課程名稱︰機率導論 課程性質︰數學系大二必修 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）︰2015/06/23 考試時限（分鐘）：110 試題 : 1. (40 points) Suppose that X is uniformly distributed on the interval [0, 1] and that, given X = x, Y is uniformly distributed on the interval [1-x, 1]. (a) (15 points) Determine the joint density f(x, y). (Be sure to specify the range.) (b) (10 points) Find the probability P(X ≧ 1/2, Y ≧ 1/2). (As usual, you can leave your answer in raw form, such as 1 - 1/e.) (c) (15 points) Find the conditional density, f (x|1/3), of X given Y = X|Y 1/3. Be sure to specify the range. 2. (40 points) Suppose X and X are independent random variables, each having 1 2 density -4 ╭ 3x , for 1 < x < ∞ f(x) = ╯ ╰ 0, otherwise (a) (20 points) Let Z = max(X , X ) denote the larger (maximum) of the two 1 2 random variables. Find the p.d.f (density) of Z. (b) (20 points) Let Y = X /X . For y ≧ 1, find P(Y ≧ y). 2 1 3. (40 points) Suppose that X and Y are independent random variables with moment -2 -3 generating function M (t) = (1 - 2t) and M (t) = (1 - 2t) . Find Var(X + X Y Y). 4. (40 points) Show that if N (t) and N (t) are independent Poisson processes 1 2 with rate λ and λ , then N(t) = N (t) + N (t) is a Poisson process with 1 2 1 2 rate λ + λ . (Hint: There are two possible approaches to this problem. The 1 2 first approach is to find P[N(t) = n], the probability mass function of the Poisson process, and the second approach is to check on the requirements of a Poisson process for N(t).) 5. (40 points) Suppose that in a certain very volatile industry, failures among companies occur at the rate of 2.5 per year. Assuming the occurrence of such business failures to be a Poisson process. (a) (20 points) Find the probability that 4 or more failures will occur next year. (b) (20 points) Find the probability that the next two failures will occur within three months of each other. 6. (45 points) A non-negative random variable X has mean 100 and variance 100. (a) (15 points) What does Markov's inequality say about P(X ≧ 400)? (b) (15 points) What does Chebyshev's inequality say about P(X ≧ 400)? (c) (15 points) Let S be the sum of n independent variables, each with the n same distribution as X. Find a sequence x so that P(S /n > 100 + x ) n n n converges to 1/4 as n → ∞. 7. (60 points) (coupon collector problem) Sample from n cards, with replacement, indefinitely, and let N be the number of cards you need to get each of n different cards are represented. (a) (30 points) Let N be the number of coupons needed to get i different i coupons after having i-1 different ones. Determine E(N ) and Var(N ). i i (b) (30 points) Write N in terms of N , ..., N . Find a sequence a such that 1 n n as n → ∞, N/a converges to 1 in probability. n 8. (55 points) (Random Incidence Paradox) Consider observing successive buses on an urban bus line as they arrive at and depart from the same bus stop. Suppose that 8 of 9 waits between successive buses are 1 minute (short gap), but that 1 in 9 each intervals equals 10 minutes (long gap). (a) (10 points) What is the probability that this passenger faces a long wait of 10 minutes for the next bus? Suppose that a second would-be passenger arrives at the same bus stop, but at a time that is random with respect to bus arrivals. (b) (15 points) It states that is the probability that this passenger faces a long wait of 10 minutes for the next bus is calculated as follows 1 ─×10 9 5 ─────── =─. 1 8 9 ─×10 + ─×1 9 9 Please explain why the above calculation is correct. (c) (10 points) Consider the first passenger who just missed the bus. Calculate his or her expected waiting time. (d) (20 points) Determine the average waiting time of the second passenger. /* 第7題是說，假設你現在在收集某個東西(e.g. 以前7-11的小磁鐵)，總共有n種，可是你在 拆開包裝以前不知道裡面是哪一種。而你的目標，就是把每個種類的東西都收集到。 */ -- 2 2 1 ψxavier13540 給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則 ╰x(0) = x° t ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三 次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。 --

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