随机过程课后习题答案

标准教材：随机过程基础及其应用/赵希人，彭秀艳编著索书号：O211.6/Z35-2备用教材：（这个非常多，内容一样一样的）工程随机过程/彭秀艳编著索书号：TB114/P50历年试题（页码对应备用教材）2007一、习题0.7（1）二、习题1.4三、例2.5.1—P80四、例2.1.2—P47五、习题2.2六、例3.2.2—P992008一、习题0.5二、习题1.4三、定理2.5.1—P76四、定理2.5.6—P80五、1、例2.5.1—P802、例2.2.2—P53六、例3.2.3—P992009（回忆版）一、习题1.12二、例2.2.3—P53三、例1.4.2与例1.5.5的融合四、定理2.5.3—P76五、习题0.8六、例3.2.22010一、习题0.4（附加条件给出两个新随机变量表达式，间接说明老师给出证法不够合理）二、例1.2.1三、例2.1.4四、例2.2.2五、习题2.6六、习题3.3引理1.3.1解法纠正许瓦兹不等式222EXYEXEY证明：2222222222200440EXYEXEXYEYEXYEXEYEXYEXEY例1.4.2解法详解已知随机过程,XttT的均值为零，相关函数为121212,,,,0attttettTa为常数。求其积分过程0,tYtXdtT的均值函数Ymt和相关函数12,Ytt。解：0Ymt不妨设12tt121222212121211220012210012121212000,,,,YtttttttttEYtYtEXdXddddddd2221211222212221122222122212121212120000220022002200222211||111111||211tttaattaaaatttaaatataattattatateddeddededaaededaatteeaaaateeeaa同理当21tt时2112112221,1attatatYttteeeaa（此处书上印刷有误）例1.5.5解法同上例1.5.6解法详解普松过程公式推导：1lim!lim1!!!1lim1!!lim1lim!limlim!第一项可看做幂级数展开：第二项将分子的阶乘进行变换：NkNNkkNNkkNNkNkqtqtNNkNkkkNNPXtkCPNqtqtkNkNqtqtNkkqteeNNNqtqtNkN!lim1!NkkkkkNkNqtNqtqtNk!1lim1!!!NkkNkqtPXtkNqtqtNkkqtek例2.1.2解法详解设,Xtt为零均值正交增量过程且2212121,EXtXttttt，令1YtXtXt，试证明,Ytt为平稳过程。证明：10YmtEYtEXtEXt1212112221121122122121121221,1191,,1111111YYttEYtYtEXtXtXtXtttttEXtXtXtXtEXtXtXtXtXtXtXtXtEXtXtXtXt此处讨论方法同第二章习题第题不妨设其他情况计算方法相同1212212121121111111EXtXtXtXtEXtXtXtXtEXtXtXtXt2121221010011EXtXttttt同理可解出其他情况，整理得：212112211,1,0,1Ytttttttt例2.1.3印刷有误2211221222220022222200214exp2112112cossin2exp2111EYtYtxxxxdxdxdd例2.2.2解法详解设,Xtt为平稳正态过程，0EXt，且XB为已知，求作用于平方滤波器时，输出过程2,YtXtt的统计性质。222222222220,000000YXYYYXXXXXXmtEYtEXtBBttEYtmtYtmtEXtBXtBEXtXtBBBEXtXtB此处利用第一章习题第5题结论222200XXXXBBBB=2+2例2.2.3解法详解coscos11cos22cos22cos22cos2cos2sin2sin2cos20sin200EXtXtEttEtEEtEtEEtEEtEt2222cos11cos11cos1112121jxjxExdxxxdxxeedxdxxx21111111jxjxssjxjjsjjedxxedxjxjxesdssjejdsss21111111jxjxssjxjjsjjedxxedxjxjxesdssjejdsss-1+12111,0,0,0,0jxsjjedxxejdssseses2111,0,0,0,0jxsjjedxxejdssseses22cos1121211,021,02jxjxEeedxdxxxeeeee11cos22cos2211022EXtXtEtEee例2.3.1解法纠正0BEXtEXtXtEXt此处老师解释为常值函数默认为BEXtXt例2.3.1，例2.3.2解法详解002jed，为狄拉克函数，为e而产生定理2.5.1印刷有误，解法详解12**111122212112212121lim21lim2120TTTTTTTTTTTTTTTTEXtdtTEXtdtXtdtTTEdjdjSdSdEXtdtTSdSdS此处出错此处出错0000210210021100021lim002TTTdSdSdSCSEXtdtmST积分中值定理与等价定理2.5.2解法详解20220XXYXXXXXYtXtXtBEYtEXtXtBBEYtYtEXtXtBXtXtBBBB,Ytt为平稳随机过程，以下步骤同定理2.5.1定理2.5.3印刷有误多处连续错误，可直接覆盖充分性：将（2.5.18）式展开，有212212,1122112101221011121021202122NkNNkkllkNllkNllNlNlMlMEXkNEXkEXlXkNNBBlkNBNBNlBlNlN定理2.5.4解法详解即证101lim0lim0NkNiBkBiN1111110101lim0,,0,,1lim11lim00kNNiKNNiiKBkKkKBkkKBkcBiNBiBiNNcN剩余步骤同定理2.5.3充分性证明。定理2.5.5解法详解记,,1,0,1,Xnn为平稳随机序列，0EXn，中心相关函数为B，记YkXkXk进一步假设,,1,0,1,Ykk为平稳随机序列，记101011ˆNNkNNkYBXkXkNBEXkXkEYkBYkNBiEYniEYniYnEYnEYniBYnB则2ˆlim0NNEBB成立的充要条件是101lim0NYNiBiN证明：必要性：“”：210210212021021202212011111ˆ1limlimNYiNiNiNiNiNNYNiBiNEYniBYnBNEYniYnBNEYkYkiEYkEYkNEYkEYkiEYkNEYkEBBBiEYkN2ˆ0NNEBB充分性：“”：