概率论与数理统计4.2-方差

4.2例1.为比较两台自动包装机的工作质量，今从甲、乙两台自动包装机所生产的产品中各抽查10包，具体数据如下甲包装机产品质量X：0.52,0.48,0.53,0.47,0.56,0.51,0.44,0.52,0.48,0.49;乙包装机产品质量Y：0.61,0.46,0.60,0.40,0.52,0.39,0.58,0.45,0.57,0.42。D(X)=E(X2)-[E(X)]2连续型离散型XdxxfXExXpXExXEXEXDkkk,)()]([,)]([})]({[)(2122方差的定义例2.已知随机变量X的分布函数如下,求D(X)。011()arcsin1211xFxxxx1+-1解:2111()'()10xfXFxx其它11112211221122212020222200()()01()()122sinsincos11sin221sin(1cos2)22111(sin2)22220xEXxfxdxdxxxEXxfxdxdxxxdxxdxdd11112211221122212020222200()()01()()122sinsincos11sin221sin(1cos2)22111(sin2)22220xEXxfxdxdxxxEXxfxdxdxxxdxxdxdd221()()(())2DXEXEX11112211221122212020222200()()01()()122sinsincos11sin221sin(1cos2)22111(sin2)22220xEXxfxdxdxxxEXxfxdxdxxxdxxdxdd11112211221122212020222200()()01()()122sinsincos11sin221sin(1cos2)22111(sin2)22220xEXxfxdxdxxxEXxfxdxdxxxdxxdxdd11112211221122212020222200()()01()()122sinsincos11sin221sin(1cos2)22111(sin2)22220xEXxfxdxdxxxEXxfxdxdxxxdxxdxdd11112211221122212020222200()()01()()122sinsincos11sin221sin(1cos2)22111(sin2)22220xEXxfxdxdxxxEXxfxdxdxxxdxxdxdd方差的性质1.D(aX+b)=a2D(X);D(aX)=a2D(X)D(b)=0D(-X)=D(X)例3.已知E(X)=2，E(X2)=6,求D(1-3X)。解:22()()[()]642DXEXEX(13)9()9218DXDX方差的性质2.若X、Y相互独立,D(X+Y)=D(X)+D(Y);)()()(,,22YDbXDabYaXDYX相互独立若niiniiinXDaXaDXXXi12121)(][,,...,,相互独立若)()()(,,YDXDYXDYX相互独立若一般地，D(X+Y)=D(X)+D(Y)+2E{[X-E(X)][Y-E(Y)]}例4.设(X,Y)的联合概率密度为求E(2X±3Y),D(2X±3Y).解:)1,0(~21)(22NXexfxX，即易见，X与Y相互独立。yxeyxfyx,,21),(2)1(22)1,1(~21)(2)1(2NYeyfyY，即E(2X+3Y)=2E(X)+3E(Y)=3E(2X-3Y)=-3D(2X±3Y)=4D(X)+9D(Y)=13设随机变量X有数学期望μ和方差σ2，则对于任给ε0,有22}|{|XP定理——切比雪夫不等式221}|{|XP即推论：D(X)=0P(X=E(X))=12222222()()()()1()(())()xxxPXfxdxfxdxDXxEXfxdx证:2222222()()()()1()(())()xxxPXfxdxfxdxDXxEXfxdx2222222()()()()1()(())()xxxPXfxdxfxdxDXxEXfxdx2222222()()()()1()(())()xxxPXfxdxfxdxDXxEXfxdx例4.设X为随机变量,已知E(X)=μ,D(X)=σ2,试用切比雪夫不等式估计P(|X-μ|≥3σ).解:222()1(3)0.111(3)99DXPX222()1(3)0.111(3)99DXPX名称概率分布期望方差0-1分布二项分布泊松分布几何分布nkppCkXPknkkn,,1,0,)1()(.1,0,)1()(1kppkXPkk,1,0,!)(kkekXPk)1(ppp)1(pnpnp21pqp,...2,1,)1()(1kppkXPk名称概率分布期望方差均匀分布指数分布正态分布其它0)(1bxaxfabxexfx,21)(222)(其它00)(xexfx212)(22abba211——X的标准化随机变量*()()XEXXDXE(X*)=0,D(X*)=1E(Xk)——X的k阶原点矩E{[(X-E(X)]k}——X的k阶中心矩E(X)——X的1阶原点矩D(X)——X的2阶中心矩