课程名称︰机率导论
课程性质︰必修
课程教师︰陈 宏
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2013/03/28
考试时限（分钟）：2:15-3:10pm
试题 :
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