概率论与数理统计浙大第四版答案-第三章

10.14101X)(XEY10)1.0,10(~BYX),4(~pBX{}2≥=Ypp{}{}{}1012=−=−=≥YPYPYpQ()()9110100101.011.01.011−⋅−−−=CC264.0=056.14)(==∴pXEiX4,3,2,1111,0=⎩⎨⎧=iXi4321XXXXX+++=)()()()()(4321XEXEXEXEXE+++={}{}{}100PPXPi+==∴()()9110100101.011.01.01−⋅+−=CC743.0={}{}264.0011==−==∴iiXPXP056.1264.04)(=×=∴XE2341234X)(XEX1234{}6437433133323213=++==∴CCCXP{}6419422233323213=++==CCCXP{}6474133332313=++⋅==CCCXP{}6414143===XP162564146473649264371)(=×+×+×+×=∴XE1625162316521691)(=×+×+×=∴XE3X()=⎭⎬⎫⎩⎨⎧−=+ixPii21121ii=1,2,…….,)(XE∑∑+∞=+∞===11)(iiiipxXE()=⋅−+iiii21211()∑+∞=+=⋅−1111iii4XX022−Pk0.40.30.3)(XE)(2XE)53(2+XE)(XE2.03.024.02−=×+×−=8.2)(2=XE4.13)53(2=+XE5.X⎪⎩⎪⎨⎧=−0,)(exxf00≤xx1XY2=2XeY2−=∫∫+∞−+∞∞−===022)(2)2(dxxedxxxfXExdxxfeeExx)()(22∫+∞∞−−−=∫∞+−==0331dxex6XY⎪⎩⎪⎨⎧=,0,12),(2yyxf10≤≤≤xy)(XE)(YE)(XYE)(22YXE+=dyydyyxfxfxX∫∫+∞∞−==0212),()(10,43≤≤xx)(XE∴544)(103=⋅==∫∫∞+∞−dxxxdxxxfX53)(=YE∫∫=dxdyyxxyfXYE),()(G21121002=⋅=∫∫dyyxydxx)(22YXE+∫∫+=Gdxdyyxfyx),()(22151612)(100222=⋅+=∫∫dxyyxdxx71X2X⎪⎩⎪⎨⎧=−,0,2)(21exxf00≤xx⎪⎩⎪⎨⎧=−,0,4)(42exxf00≤xx12)(21XXE+)32(221XXE−1X2X)(21XXE1)(21XXE+)()(21XEXE+=dxxxfdxxxf)()(21∫∫+∞∞−+∞∞−+=dxexdxexxx∫∫+∞∞−−+∞−⋅+⋅=40242434121=+=)32(221XXE−)(3)(2221XEXE−=()dxxfx∫∞+∞−⋅−×=223212dxexx∫∞+∞−−⋅−×=424321285=21X2X)(21XXE))(21XEXE=814121=×=8n1~nn1~nX)(XE⎩⎨⎧=iixi,1,0ixix01ipn11−n1nxxxLQ21,nxxxx+++=∴L2111)()()()(21=⋅=+++=∴nnxExExEXEnL9XY⎩⎨⎧=,0,2)(xxfX10x⎪⎩⎪⎨⎧=−,0,)(eyYyf0y)(YXD+XY)()()(YDXDYXD+=+∴()22)()()(XEXEXD−=dxxfxXEX)()(22∫+∞∞−=∴212102=⋅=∫xdxxdxxfxXEX)()(∫+∞∞−=322210=⋅=∫dxx()181)()()(22=−=∴XEXEXD1)(=YD)()()(YDXDYXD+=+∴1819=10.X0)(XE)(XD)(XDX*=)()(XDXEX−=0=1)(∗XE)(∗XDX*)(∗XE=−=))()((XDXEXE0)())((=−XDXEXE)(∗XD=−=))()((XDXEXD1)())((=−XDXEXD)((XEXD−()22))(())((XEXEXEXE−−−=)(0))()(2(22XDXEXXEXE=−+−=11.=1=3=0=2=1)(XE)(2XE)(YE)(2YE)(XYE)(YXD+)(YXD+()22)()(YXEYXD+−+=()()222)()(2YEXEYXYXE+−++=()())()()(2)(()(2)2222YEYEXEXEYEXYEXE++−++=61223=−++=12.(YX,)⎩⎨⎧=,0,1),(yxf10,xxy.)(XE)(YE),(YXCOVxdydyyxfxfxxX21),()(===∫∫+∞∞−−10x)(XE322)(102===∫∫∞+∞−dxxdxxxfX211),()(11=+==∫∫∫+∞∞−−dxdxdxyxfyfyyY)(YE02)(11===∫∫+∞∞−−dyydyyyfY∫∫=dxdyyxxyfXYE),()(G∫∫−=10xxxydydx=0),(YXCOV∴−=)(XYE)(XE)(YE=013.()YX,⎪⎩⎪⎨⎧+=,0),(81),(yxyxf20,20≤≤≤≤yx,)(XE)(YE),(YXCOVXYρ,)(YXD+.()4181),()(20+=+==∫∫∞+∞−xdyyxdyyxfxfX20x41)(+=yxfY20y)(XE)(YE=674)(202=+==∫∫∞+∞−dxxxdxxxfX∫∫=dxdyyxxyfXYE),()(G3482020=+=∫∫dyyxxydx),(YXCOV∴−=)(XYE)(XE)(YE361676734−=×−)((),(YXyDDYXCovXY=ρ()22)()()(XEXEXD−=dxxfxXEX)()(22∫+∞∞−=3541202=+⋅=∫dxxx3611364935)(=−=∴XD)(YD=)((),(YXyDDYXCovXY=∴ρ1113611361−=−=)(YXD+),(2)()(YXCovYDXD++=95=14.XY⎪⎩⎪⎨⎧,0,1),(πyxf122≤+yxXYXYππ211121),()(22xdydyyxfxfxxX−===∫∫∞+∞−+−−∫∞+∞−−==π212),()(ydxyxfyfY()()yxfyfxfYX≠∴XY=XYE∫∫=GdxdyyxxyfXYE),()(01111122=⋅=∫∫−−−−dyxydxxxπ)(XE)(YE=012)(211=−==∫∫∞+∞−−dxxxdxxxfXπ0=−=YEXEXYEYXCov)(2XEXD=2)(XE−=1=YD)(2XE112)(21122=−==∫∫∞+∞−−dxxxdxxfxXπ010===∴YDXDYXCovXYρXY15YX,,36)(,25)(==YDXD,4.0=XYρ)(YXD+)(YXD+)()(),(YDXDYXCovXY=ρ4.065),(=×=YXCov12),(=∴YXCov)()()(YEXEXYE−=)()()(YEXEXYE=∴)(YXD++=)(XD)(YD),(2YXCov+851223625=×++=)(YXD−+=)(XD)(YD),(2YXCov−)(YXD−∴371223625=×−+=16(),,~2σµNX(),,~2σµNYYX,,1YXZβα+=YXZβα−=2βα,)()(),(212121ZDZDZZCovZZ=ρ),(21ZZCov,(YXCovβα+=)YXβα−+αβα−=),(2XXCov),(YXCovαβ),(YXCov),(2YYCovβ−−=)(2XDα)(2YDβ−=2(α22)σβ=+=)()(1YXDZDβα+2(α22)σβ=−=)()(2YXDZDβα+2(α22)σβ22222121)()(),(21βαβαρ+−==∴ZDZDZZCovZZ1701)10.....2,1(=iXi⎭⎬⎫⎩⎨⎧∑=1016iiXP()21=iXE()121=iXD⎭⎬⎫⎩⎨⎧∴∑=1016iiXP⎭⎬⎫⎩⎨⎧≤−=∑=10161iiXP⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧−≤×−−=∑=121056121021101101iiXP137.0863.0110121=−=⎟⎟⎠⎞⎜⎜⎝⎛Φ−≈180.0510015251005.0=P100=n()05.0,100~BX1{}{45}≤=XPXP⎭⎬⎫⎩⎨⎧×−≤×××−=95.055495.005.010005.0100XP489.075.41=⎟⎟⎠⎞⎜⎜⎝⎛−Φ≈2{}{}{}410104≤−≤=≤XPXPXP489.0=1921.5100180220)100,....2,1(=iXii10021....XXXX+++={}=≤≤220180XP{}220179≤XP{}{}179220≤−≤=XPXP−⎭⎬⎫⎩⎨⎧××−≤××−=5.110021002205.11002100XP⎭⎬⎫⎩⎨⎧××−≤××−5.110021001795.11002100XP==()−Φ33.1()4.1−Φ=9192.019082.0+−=0.8274202005%90%XX~b(200,0.05)nP{Xn}0.9—:,10)(==npXE(),5.91)(=−=pnpXD}5.9105.910{}{−≤−=≤nXPnXP9.0)5.910(≥−Φ≈n14285.15.910≥≥−nn