[試題] 102-1 周雲雄 機率論一 期中考
課程名稱︰機率論一
課程性質︰研究所選修
課程教師︰周雲雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2013.11.19
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
考題分三部分，共同部分、外系研究生、數學系研究生及大學部學生
老師將學生分兩群，外系研究生及其他
外系研究生寫共同+外系研究生部分的試題
其他人寫共同+數學系研究生部分及大學部學生部分的試題
共同
1.(i )State Kolmogorov's three series theorem
(ii) ∞
Let {Xi} be iid with standard N(0,1) distribution.
1 ∞
Show that for any real number t, Y(t)=ΣXn*sin(nπt)/n converge a.e.
n=1
2.Assume {Xi} are uncoreelated with E(Xi)=μ and Var(Xi)≦C＜∞ for all i
Show that Sn/n --> μ in L2 and in probability
3.Fatou's lemma
Suppose Xn≧0 and Xn-->X in probability. Show that liminf E(Xn)≧E(X)
4.Renewal Theory
∞ ∞
Let {Xi} be iid with 0＜Xi＜∞ Let Tn=ΣXk (meaning the time of the nth
1 k=1
occurence of some event)
Define Nt=sup {n:Tn≦t}
If E(X1)=μ<∞, then show that lim Nt/t=1/μ a.e.
t->∞
外系研究生部分
5.(i )State SLLN
(ii)Prove SLLN under the extra assumption that E(X1^4)＜∞
6.2nd Borel-Cantelli Lemma
If events An are independent, then ΣP(An)=∞ implies P(An i.o.)=1
7.If {Xi} are iid with E│X1│=∞, then prove that (i) P(│Xn│≧n i.o.)=1
(ii) P(lim Sn/n exists 屬於 (-∞,∞))=0
本系學生及大學部學生
5.Prove that Xn-->X a.e. implies Xn-->X in probability.
Show by an example that the converse does not hold.
6.If events An are pairwise independent and ΣP(An)=∞.
n n
Show that (Σ1(Am))/(ΣP(Am))-->1 a.e.
m=1 m=1
1(Am)：indicator function
7.For any fixed t, is the limit Y(t)in problem 1(ii) normal?
8.Let {Xi}be iid with P(X1=±1)=0.5 (simple random walk)
(i )Is {Sn=0 i.o.} a tail-event?
(ii)Define Pn=P(Sn=0) for n≧1 and set P0=1
For n≧1 fn=P(Sn=0 but Sk≠0 for 1≦k＜n), f0=0
∞
Show that P(Sn=0 for some n≧1)=Σfk
k=1
n
(iii) Show that Pn=Σ fkP(n-k) for n≧1
∞ k=1 ∞
(iv) Let P(x)=ΣPn*X^n and F(x)=Σfk*x^k
n=0 k=0
Show that P(x)=1/(1-F(x)) for │x│＜1
∞
(v) Show that lim P(x)=∞ and thus Σfk=1
x->1 k=1
So P(Sn=0 for some n≧1)=1, that means simple random walk is recurrent!
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