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1、习题与解答3.11.一批产品中有一等品 50%,二等品 30%,三等品20%。从中有放回的抽出5件,以分别表示取出的5见中一等品、二等品的件数,求(X,Y)的联合分布列.解:这是一个三项分布,若驱除的5件中有i件一等品、j件二等品,则有5-i-j件三等品,所以当i=0,1, ,5, j=0,1, ,5, i+j5时,有 用表格形式表示如下:1.000000.002430.028350.13230.30870.360150.16807列和0.031250.000000.000000.00000.00000.000000.0312550.156250.000000.000000.00000.000
2、00.093750.0625040.312500.000000.000000.00000.11250.150000.0500030.312500.000000.000000.06750.13500.090000.0200020.156250.000000.020250.05400.05400.024000.0040010.031250.002430.008100.01080.00720.002400.000320行和543210X Yl 行和就是X的分布b(5,0.5).l 列和就是Y的分布b(5,0.3).l P(X2,Y1)=0.09+0.135+0.0675+0.15+0.1125+0.
3、09375=0.648752.100件产品中有50件一等品、30件二等品、20件三等品。从中部放回地抽取5件,以X、Y分别表示取出地5件中一等品、二等品地件数,求(X,Y)的联合分布。解:这是一个三维超几何分布,若取出的5件中有i件一等品、j件二等品,则有5-i-j件三等品,所以当i=0,1, ,5,j=0,1, ,5, i+j 5时,有用表格形式表示如下:0.318910.000000.000000.066060.141560.092740.0185520.152950.000000.018200.053930.054890.022710.0032210.028140.001890.0072
4、80.010240.006590.001930.000210行和543210X Y30.318910.000000.000000.000000.113250.156200.049460.152950.000000.000000.000000.000000.091770.06118450.028140.000000.000000.000000.000000.000000.02814列和1.000000.001890.025480.130230.316290.365350.16076l 行和就是X的分布h(5,100,50)(超几何分布)l 列和就是Y的分布h(5,100,30)(超几何分布)l
5、P(X2,Y1)=0.661583.盒子里有3个黑球、2个红球、2个白球,从中任取4个,以X表示取到黑球的个数,以Y表示取到红球的个数,试求P(X=Y).解 4.设随机变量 ,i=1,2的分布列如下,且满足P( =0)=1;试求P( ). 解 记( , )的联合分布列为 -1 0 1 P0.25 0.50.25-101-101-101-1000100(3)0100.632120.0000010.232540.13534X Y012-101/31/1201/60015/1200X-101P5/121/65/12Y012P7/121/31/12X Y01200.160.320.1610.080.1
6、60.0820.010.020.01X Yy1y2y3x1a1/9Cx21/9b1/3 YX y1y2y3P(X=xi)x1a1/9ca+c+1/9x21/9b1/3b+4/9P(Y=yj)a+1/9b+1/9c+1/31X Y12300.050.150.2010.070.110.2220.040.070.09U123P0.120.370.51V012P0.400.440.16X-101P1/41/21/4Y01P1/21/2X Y01P(X=i)-1p11p121/40p21p221/21p31p321/4P(Y=j)1/21/21X Y01P(X=i)-1p1101/40p21p221/21p3101/4P(Y=j)1/21/21X Y01-11/40001/211/40Z01P1/43/4X-101P0.13440.73120.1344