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）︰2013/03/28 考試時限（分鐘）：2:15-3:10pm 試題 : Introductory Probability Quiz 1 Thursday 2:15-3:10pm, March 28th, 2013 1. (20points) Let X_n have a geometric distribution with p = λ/n in which P(X = x) = p(1-p)^x an x = 0,1,….. (a) (10 points) Compute P(X_n/n > x). (b) (10 points) determine lim P(X_n/n > x) in terms of λ. n→∞ 2. (20 points) The symmetric difference between two events, A and B say, is c c defined to be A△B = (A ∩B)∪(A∩B ). Show that P(A△B)=P(A)+P(B)-2P(A∩B). 3. (25 points) Prove the following version of Stirling's n! formula: lim ────────── exists ane it is a finite number. n→∞ n^[(n+1)/2]exp(-n) n! Write d_n = log(──────────). n^[(n+1)/2]exp(-n) (a) (15 points) Compute d_n - d_n+1 and show that 0 < d_n - d_n+1 < 1/12n - 1/[12(n+1)]. Hint: The series expansion of the function log[(1+t)/(1-t)] near t=0 might help. (b) (10 points) Show that the lim d_n exists and finite. n→∞ 4. (20 points) Let X be a uniformly chosen number from the set {1,2,3} and Y is an independent random number uniformly chosen from {1,2}. (a) (10 points) Find the distribution of Z = XY. (i.e. find the probability mass function.) (b) (10 points) Find the distribution of W = cos(2πZ/3). 5. (15 points) An insurance company insures 3000 people, each of whom has a 1/1000 chance of an accident in one year. Use the Poisson approximation to compute the probability there will be at most 2 accidents. 6. (20 points) Let A_1,A_2,…,A_n be events. Show that n Σ P(A_i) - Σ P(A_i∩A_j) + Σ P(A_i∩A_j∩A_k) i=1 1≦i<j≦n 1≦i<j<k≦n n n ≧ P(∪ A_i) ≧ Σ A_i - Σ P(A_i∩A_j). i=1 i=1 1≦i<j≦n --

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