《概率论期末复习资料》习题三龙永红版答案.doc

****题三答案 P84~1. 证明: ???????? 11122122,,,,FyxFyxFyxFyx??????? 121221,xP,xPyyx<xyyx<x???????? 0,xP 2121????y<yyx<x 由概率的非负性,知上式大于等于零,故得正. 4. 解: ①, 2X 1X 01?? jxXP 22? 012 1/56 10/56 10/56 5/56 20/56 10/56 3/28 15/28 5/14 ?? ixXP 11? 3/8 5/81 ②.?????? 0}0|0{10,0XP 11221???????xPxxPX14 556 20 8 321 20 8 311 27???????????????C?????? 1,10,0XP 212121???????xxPxxPX56 21 ????????? 2,11,11010xP 21212121???????????xxPxxPxxPx28 13 56 26 56 10 14 51????? 5. 解: ①由概率的性质?? 12 1, 1 43100?????????????????????? k dxdy ek dxdy yxf yx 又?? 21 1, 2 4320000???????????????? k dxdy ek dxdy yxg yxxx②?? xxyxXeekdy ekxf 331 431034 1 ??????????? 0x> ?? yyyxYeekdx ekyf 441 431043 1 ??????????? 0y> 当y>0时???? xxyxxXeedx ekxg 7343204 21 ???????? 0x> 当x>0时?? yyxyYedx ekyg 74327 ???????? 0y> 4. 解: ①????????????其他, ,0 1020,2 1, yxyxf ???????????????其他, ,0 1020,2 2/1, 00yx xy dxdy yxF yx②?? 2 12 1 10??? dy xf X20??x ;?? 12 1 20??? dx yf Y ,10??y ??????????0 22 1 0x dx xF xX 其他 20??x ;????????0 1 0y dx yF yY 其它 10??y ③?? 3 22 112 1x< yP 110x< y 2 2 2???????? dxdy dxdy x 5.??????0 2,yxf 其它 10???yx???????????? 0 122 1x dy xf x X 其它 10??x???????? 0 22 0y dx yf yY 其它 10??y 6.①?????????????????????????2,2,1 21,2,1 2,20,2 21,20,2 1 ,0,y> x yx> y y> x x yx yxyxF 其它????????2 1 0,yxf21,20????yx 其它②???? yYxXPyxG??? 22,,?? y<Yyx<XxP?????, 讨论如下: ????????????????????????4,4,1 41,4,1 4,402 1,00 41,40,2/)(),(yx> yx> y yx x yx yxx xyyxG, 或, ③??????2 3,y> x< P1 =4/12 12 1 22/3 10?????? dxdy dxdy D 7. 证明: ???? 2121y<YyPx<XxP?????????????? 1212yFyFxFxF yyxx??????????? 1121,12,22,,,,,yxFyxFyxFyxF yyxyxyx?????? 2121,y<Yyx<XxP???故独立得证. | 21?xx 01 P3 13 2 9. 解: ①ij xi ijPPP?②?? i j xii jppyyP???? 1 ③i j xii i j xijipp ppP?? 10. 解: ①4 1 11?P0 12?P4 1 13?P2 1 22?P ②???? 014 1 11?????xpypP 故不独立. 11. xy1y 2y 3y xip 1x 2x yjp 1/24 1/8 1/6 1/8 3/8 1/2 1/12 1/4 1/3 1/4 3/4 1 12. 证明:必要性: 由:n n nm mn mm n nppp ppp ppppppppp ppp ppp,,, ,,, ,,,,,,, ,,, ,,, 21 21 21 21 2 22 21 1 12 11????????????? 2 1 独立? 1 秩为?充分性,若 nm mm n n mn mm n nppp