概率论与数理统计(经管类)第四章课后习题答案

1、0000习题4.11设随机变量X的概率密度为(Ifx2x,0 x1,0,其他;(2fxe11，ococ求E(X解:(1EXxfxdxco2xdx2-10(2EXxfxdxxe|co000ooO2.设连续型随机变量X的分布函数为Fx0,x1,ab-aicsiiix,1x1丄x1.jaixs的挣数为7试确定常数ab并求E(X.解(1fxFx,1X10,其他fxdxb、hxdxb-aicsnix111,即b17T又因当1X1时FXfxdx17T171Xdx17i-arcsiiixx1X17-arcsinx1,g31(2EXxfxdx兀.oo03设轮船横向摇摆的随机振幅X的概率密度为fxloexOQx

2、0.求E(X.解:EXxfxdxcoeodx设X1,X2,.Xn独立同分布,均值为比旦设YXX，求E(Y.解EYEXXEXnpM设(X、Y的概率密度为fx,ve,0 xl,y0,0.0.0.0.0淇他.求E(x+Y解:EXYxyfx,ydxdyoooxyedxdyoey-edy设随机变量XI,X2相互独立，旦XI,X2的概率密度分别为fx2e,x0,fx3ex0,x0,132132132132求：1E2X3X;2E2X3X;3EXX解:E2X3X2EX3EX2322E2X3X2EX3EX13Xoc3edx13Xocde13xeoOeoodx130e2xcodxe3xcodx3113EXXEXE

3、X求E(X.解:EXX&p00.100.310.210.120.120.20.9&设随机变量X的概率密度为fxexa.Ox1,0,其他.且E(X=0.75,求常数c和a.解:EXxfxdxxcxadx0.75000习题4.21.设离散型随机变量X的分布律为X100.512P0.10.50.10.10.2求EX,EX,DX.解EX10.100.50.50.110.120.20.45EX10.100.50.50.110.120.21.025DX10.450.100.450.50.50.450.110.450.120.450.20.82252.盒中有5个球、其中有3个白球,2个黑球，从中任取两个球，

4、求白球数X的期望和方差.解:X的可能取值为0,1,2PX0CC0.1|e_，X，dXPX1C0.6PX20.3EXO0.110.620.31.2EXEXEXEXEXEXDXO1.20.111.20.621.20.30.1440.0240.1920.363.设随机变量X.Y相互独立，他们的概率密度分别为fXx2e、x0.0,x0,fYy4,0o其他，求D(X+Y.解DXYDXDY4.设随机变量X的概率密度为fXx此为奇函数故=o穷带入结果都样故求D(X解:EXe|dx2e11dx2x2exe2DX=EXEX25.设随机变量X与Y相互独立，且D(X=1,D(Y=2,求D(XY.解:DXYDXDY1

5、236.若连续型随机变量X的概率密度为fxaxbxc.O1、0,其他，且E(X=0.5,D(X=0.15.求常数a,b,c.解：EXxaxbxcdxa4b3c20.5axbxcdxa5b4c0.150.50.4fxdxax2bxc10dxa3b2c1解得a=12,b=-12,c=3.习题4.31.设两个随机变量XY相互独立方差分别为4和2,则随机变量3X-2Y的方差是D.A.8B.16C.28D.442设二维随机变量（X.Y的概率密度为fxy1(a+b+cx)dx(a+b+cx)dx8xv,0 x2,0y2,求Cov(X,Y.解】EXx8xvdydxx8yx8-y220dx76EYy8xydx

6、dv76EXYxy8xvdydx43CovX,YEXYEXEY47713设二维随机变量（XX的概率密度为fx,vye,x0,0,0,其他求X与Y的相关系数pxy.解:EXxyedycocodx1EYyedxocdyyeedxocdyyeoodyyooyeoO:运用分部积分法.Mreoody0e2yocdv2eyoodv2EXYxyedycodx2CovX,YEXYEXEY2210所以pxyCovX,YDXDYO设二维随机变量(X,Y服从二维正态分布，且E(X=O,E(Y=O,D(X=16,D(Y=25,Cov(X,Y=12,求(X,Y的联合概率密度函数f(x,y.解：fx,vHOOepgopp

7、卩ooMoEX0,EY0pl0,p20,DX16,DY25ol4,o25CovX,Y12pCovX,YDXDY123fx,y13271e2532x2163xy50y225证明D(XY=D(X+D(Y2Cov(X,Y.证：DXYEXYEXYEXEXYEYEXEX2EXEXEYEYEYEYDXDY2CovX,Y设(X,Y的协方差矩阵为C4339，求X与Y的相关系数pxy.解:C4339CovX,Y3,DX4,DY9pxyCovX,YDXDY31自测题4、选择题1设随机变量X服从参数为0.5的指数分布，则下列各项中正确的是B.E(X=0.5,D(X=0.25E(X=2,D(X=4E(X=0.5,D(

8、X=4E(X=2,D(X=0.25解:指数分布的EXX,DXX设随机变量X,Y相互独立，且XB(16,0.5,Y服从参数为9的泊松分布，则D(X2丫十1=C.TOC o 1-5 h z14134041解:DXnpq160.50.54,DYX9DX2Y1DX4DYD1449040已知D(X=25,D(Y=1,pxy=0.4,则D(X-Y=B.6223046设（X、Y为二维连续随机变量、则X与Y不相关的充分必要条件是C.X与Y相互独立E(X+Y=E(X+E(YE(XY=E(XE(Y(X,YN(p41QQ,0解:X与Y不相关pxy0,CovX,Y0EXYEXEY设二维随机变量(X,YN(1丄4,9,

9、，则Cov(X,Y=B.A.TOC o 1-5 h z31836解：pxy12CovX,YDXDYCovX,Y23,CovX,Y3已知随机变量X与Y相互独立，且它们分别在区间-1,3和2,4上服从均匀分布，则E(XY=A.3TOC o 1-5 h z61012解XU1,3,YU2,4EXabl31,EY243EXYEXEY133设二维随机变量(X,YN(O,O,l,l,O,0(x为标准正态分布函数，则下列结论中错误的是C.X与Y都服从N(0,l正态分布X与Y相互独立Cov(X,Y=l(X,Y的分布函数是xQy二、填空题1.若二维随机变量(X,YN，卩qQ,0,且X与Y相互独立，则p=0.解:C

10、ov(X,Y=0设随机变量X的分布律为3.X-1012P0.10.20.30.4令Y=2X+1,则E(Y=3.解:E(2X+1=(2*-1十1*0.1十(2*0十1*0.2十(2*1十1*0.3十(2*2十1*0.4=3已知随机变量X服从泊松分布，且D(X=1,则PX=1=e.解:DXA.1PX1Xek1!e4.设随机变量X与Y相互独立、且D(X=D(Y=1,则D(XY=2.已知随机变量X服从参数为2的泊松分布,EX=6.解:EXA.2QXA.2,EXEXDX426设X为随机变量，且E(X=2,D(X=4,则EX=8.已知随机变量X的分布函数为Fx0,xOx4,0 x41,x4则E(X=2.解

11、:fxFx,0 x40,其他EXx440dx0&设随机变量X与Y相互独立，且D(X=2,D(Y=1,则D(X2Y十3=6.三、设随机变量X的概率密度函数为fx32x,1x1,0,其他试求:(1E(X,D(X;(2P|XEX|2.解：(1EXxdxODXEXEX32x32x51135(2P|XEX|2P|X|fxdxxdx1四、设随机变量X的概率密度为fxxOx12x,1x20,其他试求:(1E(X,D(X;(2EX淇中11为正整数.解：(lEXxdxx2xdxlDXEXEXxdxx22x2EXnxdxxn2x五、设随机变量X1与X2相互独立，且X1N(w,X2Nq.令

12、X=X1+X2,Y=X1X2.求:(1D(X,D(Y;(2X与Y的相关系数pxy.解：(1DXDX1X2DX1DX2oo2oDYDX1X2DX1DX22O(2CovX,YEXYEXEY0pxyCovX,YDXDY0六、设随机变量X的概率密度为fx2e,x0,0,x0.(1求E(X,D(X;(2令YXEXDX,求Y的概率密度fY(y.解(1EX2xe2xdx12DXEXEX2x2e2xdx14121414(2YXEXDXX12122X1由Y=2X1得XY12,X，=fYy2eY12,Y1200,Y120=e,v10,y1七、设二维随机变量(X,Y的概率密度为fx,v2,Ox1,0y,0,其他求i(1E(X+Y;(2E(XY;(3PXY1.解:(1EXYdx2xvdv2xxdx1(2EXYdx2xvdvxdx(3