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08~09-1学期《概率论与数理统计》试题A参考答案 一、填空题:1、2;2、0.4;3.?21?,??;4、2.6;5、?2(n) 99二、选择题:1、C;2、D;3、B;4、B;5、C 三、解:设Bi=“取出的零件由第 i 台加工”(i?1,2)
21P?A??P?B1?P?AB1??P?B2?P?AB2???0.97??0.98?0.973
3313?1?1?1??1?P?X?0,Y?3?????,P?X?1,Y?1??C3?????,
88?2??2??2?32四、解:由题意知,X的可能取值为:0,1,2,3;Y的可能取值为:1,3. 且
P?X?2,Y?1??C231?1??1?3?1??????,P?X?3,Y?3?????.
8?2??2?8?2? Y X 0 1 2 3 23于是,(1)(X,Y)的联合分布为 1 0 3 1 80 0 3 83 80 1 8(2)P?Y?X??P?X?0,Y?3??1
8五、解:随机变量
X的密度函数为
f?x??设随机变量Y的分布函数为FY
12?e?x22
????x????
FY?y??P?Y?y??PX2?1?y?PX2?y?1 ①. 如果y?1?0,即y?1,则有FY?y??0; ②. 如果y?1,则有
??y?,则有
???
FY?y??P?X2?y?1??P?y?1?X?y?112?y?1??
??y?1?e?x22dx??x2222?y?1?0e?x22dx
?2?即FY?y???2???所以,
y?1?0edxy?1y?1
0?1?2?y21e??fY?y??FY??y???2?2y?1?0?y?1y?1
y?1??1e2y?1?即 fY?y???2?y?1.
?0y?1???1?xdx?0 六、解: ① E(X)??xe??2D(X)?E(X2)?[E(X)]2
????1?x1??x2edx?0?2?x2e?xdx?2
??022②Cov(X,所以
X)?E(XX)?E(X)E(X)??不相关.
?????????????xx1?xedx?0?0 2X与X七、(本题满分10分) 解:(1)由1??f(x,y)dxdy??0?????0??Ae?(x?2y)dxdy
?A?0e?xdx?0e?2ydy???1A 所以A?2 2?e?x x?0(2)X的边缘密度函数:fX(x)??f(x,y)dy??
??其他?0,???2e?2y y?0Y的边缘密度函数:fY(y)??f(x,y)dx??
??其他?0,??f(x,y)?fX(x)fY(y),所以X,Y是独立的
2八、解:⑴. 当??0为未知,而???????(3)因
为已知参数时,似然函数为
?1n2?L??2??exp??2??xi????
?2?i?1?n1n222??x??因而 lnL?????ln?2???? ?i22?2i?1?n1n122所以 lnL?????2???xi????4?0
2i?12?????2????2n2?2?1n2解得????xi???ni?122
21n????Xi???2. 因此,?的极大似然估计量为?ni?12,2,?,n?, ⑵. 因为Xi~N??,?? ?i?1所以所以
Xi??E?Xi????0,D?Xi?????2 ?i?1,2,?,n?,
?~N?0,1? ?i?1,2,?,n?,
2所以E??Xi?????E?Xi???