1、马春光马春光machunguanghrbeu.edu 哈尔滨工程大学=U1nnA 显然，如果 F F 是一事件域，那么 (1)F F；(2)若A,B F，则A-B F F；(3)若An F F，n=1,2,则 F F.=I1nnA11()()nnnnPAP A()(|)P ABP B AA不难验证，条件概率P(|A)符合定义1.1.2中的三个条件，即(1)F ，P(B|A)0；(2)P(|A)=1；(3)设BnF ，n=1,2,BiBj=,ij,i=1,2,则 B11(|)(|)nnnnPBAP BA12121312121()()(|)(|)(|)nnnP A AAP A P AA P AA
2、AP AA AA1,iiBA1()()(|)iiiP AP B P A B1,iiBA1()(|)(|),1,2,()(|)iiijjjP B P A BP BAiP B P A B1212()()()()kkiiiiiiP A AAP A P AP ABAABB,B B C C BC BC CB BC(),(),CBBCBCBCBC,BC BC BC 12,nA AA11112()1()1()1()()()nniiiiniinPAPAPAP A P AP A ,|()xRXx Fx xR()lim()0,()lim()1xxFF xFF x ()0,()1FF (),iixxF xP xR(
3、)(),xF xf t dt xR12()(),(),(),nXXXX 12(,),nx xxR x1122|(),(),()nnXx XxXxF12112212()(,)(,)(,)nnnnnF xF x xxP Xx XxXxxx xxR111(,)0,1,2,iinF xxxxin(,)1F 1212111121111112(,)(,)(,)(1)(,)0nniiiniiiijjjnij nnnF y yyF y yyx yyF y yyx yyxyyF x xx 1122121122(,)(,)nnnnnxu xuxuF u uuP Xx XxXx 12121212(,)(,)nuuu
4、nnnF u uuf x xx dx dxd 12121212(,)=(,)kkxxxnnF x xxf y yy dy dydy 12121(,)(,)knkng x xxf x xx dxdx 1212(),(),()nXXXnFxFxFx1212(),(),()nXXXnfxfxfx1212(),(),()nXXXnFxFxFx1212()()()nXXXnFx FxFx12121212(,)()()(),nnXXXnnf x xxfxfxfxx xxR1121212212(,)(,)(,)nnnnny x xxyyx xxyyx xxy,1,2,iijjxyi jnyx1212(,)(
5、,)nnx xxJy yy1211221212(,)(,),(,),(,)YnXnnnnf y yyfx y yyx y yyx y yyJ()()1112()()2212()()12(,)(,)(,)llnllnllnnnxxy yyxxy yyxxy yy12()()()11221212(,)(,),(,),(,)YnlllXnnnnlfy yyfxy yyxy yyxy yyJ()(),()0,XYfh yh yyIfyyI1122()()()(),()0,XXYfh yh yfh yh yyIfyyI,tan2 2XUYX,y 21(),1h yyy 211(),1Yfyyy 12()
6、,0,(),0h yy yhyy y 1211(),0,(),022h yyh yyyy 12211221()()()(),0()20,0yXXYfh yh yfhyhyyeyfyy222222221(0,),(),21(0,),(),2xXyYXNfxexYNfyey 2221()221(,)()(),2xyXYf x yfx fyexy 22arctanzxyyxcos,0,sinxzzyzcossin(,)sincos(,)xxzx yzJzyyzzz2222(,)(cos,sin),0,2zzg zf zzJez2222222222220,0()(,)20,01,()(,)220,zz
7、Zzzzedezgzg zdzzedzgg zdz其它(,)1maxkn=V11()()()nkkkkSfg xg x1001limlim()()()nkkkkSfg xg x()()baf x dg xlim()()abbf x dg xa()()f x dg x()()()(0)(0)kkkkf x dg xf xg xg x()()()()f x dg xf x g x dx()cos()sin()jtxe dg xtxdg xjtxdg x()xdF x()defEXxdF xiiiEXx p()EXxf x dx()()()EYE g Xg x dF x()()g x dF x121
8、2(,)(,)nng x xx dF x xx 121212(,)(,)(,)nnnE g XXXg x xx dF x xx 2()defDXE XEXcov(,)()()defX YE XEX YEYcov(,)defXYX YDXDY22()()()DXe XEXxEXdF x2()iiiDXxEXp2()()DXxEXf x dxcov(,)()()(,)X YxEXyEY dF x y cov(,)()()iiijijX YxEXyEY pcov(,)()()(,),X YxEXyEY f x y dx dy(lim)limlim(lim)nnnnnnnnEXEXEXEX0,()rr
9、E XP X1()()()()rrrrrrxxxE xP XdF xdF xx dF x()rrE XP X()rrE XP X2()DXP XEX111()2cov(,)nniiijiiij nDXDXXX 1,0,iX1niiXX11(1),(0)iinP XP Xnn1,1,2,iEXinn11()1nniiiiEXEXEX21,iEXn2222111(),1,2,iinDXEXEXinnnn221(,)(1,1)(1)(1|1)=(1)111cov(,)()(1)(1)ijijijiijijijE XXP XXP XP XXn nXXE X XEX EXn nnnn111222=()2
10、cov(,)1121(1)nniiijiiij nnDX DXDXXXnnCnnn 12(,)defnEXEX EXEX111212122212cov(,)cov(,)cov(,)cov(,)cov(,)cov(,)cov(,)cov(,)cov(,)ndefnnnnnXXXXXXXXXXXXBXXXXXX111121cov(,)()()()0nnnnlkl kllkkl klklknkkkkXX t tE XEXXEXt tEXEX t11cov(,)nnlkl klkXX t t,1defZXjY jdefEZEXjEY()(),defjtXjtXtE eedF xt ()cossin,t
11、EtXjEtXt ()ijtxjtXiitE eep()()jtXjtxtE eef x dx()()1jtXjtxE eedF xdF x ()jtXjtctE ee()kkn knP XkC p q()jttpeq0()()njtXjtkkkn kjtnnktE ee C p qpeq例例1.4.2 设XB(n,p)即 k=0,1,2,n,0p|.(|)(|),idefijX YjxxjpFx yP Xx Yyxp=-+|.(|)(|),idefijY XjyyipFy xP Yy Xxyp=-+(,)(|)()P Xx yYyP Xx yYyP yYyeeeeee-+-+=-+00(,)
12、lim(|)lim()P Xx yYyP Xx yYyP yYyeeeeeeee+-+-+=-+|0000(,)(|)lim()(,)(,)=lim()()(,)(,)lim2=()()lim2(,)/()/X YYYYYYP Xx yYyFx yP yYyF x yF x yFyFyF x yF x yFyFyF x yydFydyeeeeeeeeeeeeeeeeee+-+=-+-+-+-+-抖=|(,)(,)(|)()()xxX YYYf u y duf u yFx ydufyfy-=|(,)(|)()defX YYf x yfx yfy=|(,)(|)()defY XXf x yfy x
13、fx=|(|)(|)defX YE XyxdFx y+-=|(|)(|)defY XE Y xydFy x+-=|(|)(|)(|)(|)ii jiijiijjijjijjE Xyx px P Xx YyE Y xy py P YyXx=邋邋|(|)(|)(|)(|)X YY XE Xyxfx y dxE Y xyfy x dy+-+-=111|,111(|,)iiinX XXXXiiinFxxxxx1111211de|,111(|,)(|,)iiiniiinfX XXXXiinE Xx xxxxxdFx xxxx1113(5)(7)333EXEX 1111111111111()()|)(|)()(|)()()()()()NNiiiiNininininininnEXE EXNEXNn P NnEXNn P NnEX P NnnEX P NnEXnP NnEX EN
