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! --

※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 140.112.217.22