随机过程答案-西交大

【第一章】1.1证明：∵1111,,,,,AFFFF且∴1F是事件域。∵222,,,,cAAFFAFAA∴22222,,,,ccAFAFAFAFAF且2,ccAAAAF∴2F是事件域。且12FF。∵2∴3F∴3F是事件域。且23FF∴123,,FFF皆为事件域且123FFF。1.2一次投掷三颗均匀骰子可能出现的点数ω为,,,,,,,,16,6,6ijkiRjRkRjikjijk∴样本空间61=,,nijikjijk事件,,|,,ijkAijk，,,,,,6,16,6iRjRkRjikjijk事件域2F概率测度,,1P677ijkAij，,,,,,16,6,6iRjRkRjikjijk则,,FP为所求的概率空间。1.3证明：（1）由公理可知0P（2）有概率测度的可列可加性可得11nnkkkkPAPA（3）∵,,ABFAB∴BAF,ABA由概率测度的可列可加性可得：()()()PBPABAPAPBA即PBAPBPA有概率测度的非负性可得0PBPAPBA，即PBPA（4）若B,由（3）则有1PAPA（5）∵121212PAAPAPAPAA假设11211111mmmkkijijkmkijmijkmkPAPAPAAPAAAPAAA成立，则11111111111111211111+1mmmmkkmmkmkkkkkmmkijijkkijmijkmmmmmkkmkijikijmPAPAAPAPAPAAPAPAPAAPAAAPAAAPAAPAPAAPAA1121111121111212111111111njkmijkmmijmijkmmmijmijkmmmkijijkmkijmijkmAPAAAPAAAPAAAAPAAAAPAPAAPAAAPAAA也成立由数学归纳法可知11211111nnnkkijijknkijnijknkPAPAPAAPAAAPAAA111122212123231231nnnnkkkkkkkknnnkkkkkknkknkkPAPAAPAPAPAAPAPAPAPAAPAAPAPAPAPA1.4（1）21040114PABPAPBPABPABPAPBPABPAPBPAPABPAPBPABPAPABPAPAPA（2）if=1elseif=PABPBCPABPBCPABPACPABCPABCPABPBCPACPABCPABCPBCPABCPABPBCPABPBC可由这个式子的轮换对称性证明这种情况（3）11111111111nnkkkknnnnkkkkkkkknkknkkAAAAPAPAPAPAnPAPAnPAPAPAn1.5!!knkkAnPXknnk，∴!11!knFXPXxPXxnk1.6由全概率公式100112211110101=1424PYXPYPXPYPXPYPXPYPYPYe1.7证明：显然111111122,,,,,,0nnnnnFxxFxxFyxPxXyxXxX假设121111222,,,,,,,0iniiiiinnFxxPxXyxXyxXyxXxX成立从而12+11111222111112221111122211122,,,,,,,,,,,,,,,,,,,0iiniiiiinniiiiinniiiiinnFxxPxXyxXyxXyxXxXPxXyxXyxXyyXxXPxXyxXyxXyxXxX（分布函数对于每一变元单调不减）也成立由数学归纳法可知121111222,,,,0nnnnnFxxPxXyxXyxXy1.8''''''',,0','xyxyxxyxyxyxyxyxxyyhxyeehxyeeeeeeeexxyy所以h是二元单调不减函数。,,0,0hxyhxyxy但对每一个变元是单调减的。,0or10,,0or10,','+,',,'','=1,=0',=1,'=1,0,xyxyxygxygxygxygxygxygxygxygxygxygxygxygxy对于每一变元单调不减；当，，，时是单调不增的。1.91323122223332222222131232231011121XXXXEYEEXEEXEXXXXXXXXXDYEYEYEYEX1.10101:1/11/1/11/11+21niiininnnpnEXpipppipppnp1.11+++0gxxdFxgxxudFxgxxudFxgxudFxgEx1.12（1）~exp,,1,0min,,0|0|01,0,01When,||1,|xxXXxYXtxYxtXfxeFxexYXttefyXtfxxtefyXteFyXteXtYtPYtXtPXtXtPYtXtPYtXtPXte（2）|xxtXtefxxtee1.13（1）12112112100|therightsideoftheequationleftside=rightsidennnPABBBPAPABBBPAPBBBAPA（2）1111||||||kkkkkkkikPAAPBAAPABAPABAPBAAPBA1.14证明：||||||||PABCPABCPBCPABCPACPBCPACPABC证毕1.1511|1,1112|1|15xXXfxxefxxxFEXXxfxxdx1.16,|||||,CovXEYXEXEXEYXEEYXEEXYXEXEYXXEYEXEYEXYEXEYCovXY1.17|,21|211XYXYYfxyfxyfyyy1|1||1,0112XYyyEXyxfxydxxydxyy|,1|XYYXXfxyfyxfxx|00||dy=2xxYXyxEYxyfyxdyx1.18证明2222222|||2||||2||||||DXYEXEXYYEXXEXYEXYYEXYEXYEXYEEXYYEXYEXY222222||||||EDXYDEXYEEXYEXYEEXYEEXYEXEXDX1.1910101,01,1XzXYXxyzzyXXZXXfyyfxyfxFZzFXYzfxdxdydyfxdxFzydyfzFzFz1.20（1）|00,01|11,120,0111,121,011,12PXYzPXYzYPYzPXYzYPYzPXzPYzPXzPYzzpzzpz（2）1,001PXYzpzpzz1.2111211211211211121212121112=2XEXXEXXXXEXXEEXXXXXEXXEXXEXXXX1.22（1）1212221212211211222121212211,exp2211+,221111,exp+4222XXYYfxxxxXYYXYYfyyyyyy（2）122221221212,12cov,2YYXDYDYYYDYDY1.23（1）2312321122221223321121232312312322282,,8yYYYxyxyyxyyyJyyyyyyyfyyyyyye（2）2212211222211212128,8yYYxyxyyJyyfyyyye1.24（1）0000001|!|!1kkckPXkekPXkPXkdFecedkcc？未算，请验证（2）00000|!|kPXkekPXkPXkdF1.25000|1|01!0nnnnmmmnnnnnmmmnnGzpzPYmXnCppPYmPYmXnpGCppnnm001=10nmmmmYnnmnGzCpppzGpzpnm1.261101143