主要内 容(0.5学时)相互独立的正态随机变量线性组合相互独立的正态随机变量线性组合第二节正态随机变量的线性组合 ( )t( )y f()X Y\uf02bYXffty dy\uf02b\uf0a5\uf02d\uf0a5\uf03d\uf02d\uf0f2\uf02b\uf02b22. (0,), (0,1),,,(0,1)YNXNX YXYN\uf073\uf073引 理设且相互独立 则 \ue000\ue000\ue000,,( ),x( ),y( )tXYX Y\uf02bX Y XYfff\uf02b证: 设的概率密度分别为222yy2221[2(() ]) ]yytt\uf0a5\uf0a5\uf0f2\uf0f2++()2221\uf070\uf07312\uf0702t y\uf02deedy\uf073\uf0a5\uf02d\uf02d\uf0a5\uf03d\uf0f2+-12[12\uf070\uf073yedy\uf073\uf02d\uf02b\uf02d\uf0a5\uf03d-222(1)212\uf0701 f( )ttX Y\uf02be\uf073\uf02b\uf073\uf02b\uf02d\uf03d(略中间过程, P100)\uf02b\uf02b2(0,1)XYN\uf073即 \ue000 1221222 X()+XYY\uf02d\uf02d\uf02b\uf03d\uf02b\uf02b\uf06d\uf06d\uf073\uf06d\uf06d\uf073\uf073证:22XN( , ) \uf06d \uf073(,() )Y aXbN ab a\uf06d\uf073\uf03d\uf02b\uf02b\uf0de\ue000\ue000211221111(0(0,))XXXNN\uf02d \uf06d\uf06d\uf073\uf06d\uf06d\uf073\uf073\uf073\uf0732122122211221. (\uf06d \uf073,), (\uf06d \uf073,),,,(\uf06d,\uf073)THXNYNXXYNY\uf02b\uf02b\uf02b\ue000\ue000\ue000\uf06d\uf073设且相互独立 则 22(0 1)(0,1)YNN\uf02d \uf06d\uf06d\uf073222 \uf0de\uf03d\uf02d\ue000\uf0732\ue00021221222(0, 1)XYN\uf02d\uf02d\uf02b\uf02b\ue000\uf06d\uf06d\uf073\uf073\uf073\uf073212222212 X[\uf073*0, (1\uf073)]YN\uf02b\uf02b\uf02b\uf02b\ue000\uf073\uf073\uf06d\uf06d212212(\uf06d,)N\uf03d\uf02b\uf02b\uf06d \uf073\uf073 22XN( , ) \uf06d \uf073(,() )Y aXbN ab a\uf06d\uf073\uf0de\uf03d1\uf02b\uf02b\ue000\ue00021\uf073212n\uf0732n111\uf06d12i12122. ,,...,,(\uf06d \uf073,)(1,2,....(...., ),n0,,...,,.,...)niinnn\uf06dnnTHX XXXNiCUC XC XN CCCCCC\uf03d\uf02b\uf02b\uf02b\uf02b\uf03d\uf02b\uf02b\ue000\ue000设相互独立 且对于任意不全为 的常数有12 C...nCCnn\uf03d\uf03d\uf03d\uf03d取\uf03d\uf03d\uf03d\uf02d\uf0e5i2122121. ,,...,,( ,\uf06d \uf073)1(1,2,..., ), (0, 1) X,,..( , )\uf06d./,,nninX XXNinXXNXnNnXnXX\uf06d\uf073\uf073