12?111??1??P?A??P?B??P?AB???1??????1??
33?4612??? P?X?1,Y?0??P?AB??P?A??P?AB?
P?X?1,Y?1??P?AB?? 即 Y X 0 1 P?X?0,Y?1??PAB?P?B??P?AB??111?? 612121 120 1 21 31211 612 ?e?y,y?04. 解：由题意知Y的密度函数为fY?y???，?X1,X2?的可能
?0, 其他取值为?0,0?，?0,1?，?1,0?，?1,1?，则?X1,X2?的联合分布律为
P?X1?0,X2?0??P?Y?1,Y?2??P?Y?1???e?ydy?1?e?1
01P?X1?0,X2?1??P?Y?1,Y?2??P????0
P?X1?1,X2?0??P?Y?1,Y?2??P?1?Y?2???e?ydy?e?1?e?212P?X1?1,X2?1??P?Y?1,Y?2??P?Y?2???e?ydy?e?2，即：
2?? X2 0 1 X1 0 ?1 1?e 0 1
e?e e ?1?2?211
5. 解：?1?由题意记区域G的面积为A?G?,则A?G??所以
1x?xdx?，???6120??6,f?x,y?????0,?x,y??G
?x,y??G?2? 关于X的边缘密度函数为
fX?x???????x2???x26dy?6x?6x,0?x?1 f?x,y?dy????0, 其他关于Y的边缘密度函数为
fY?y????????y6dx?6y?y,0?y?1?f?x,y?dx???y
?其他?0, ???3? 不独立. 因为当0?x?1,0?y?1时f?x,y??fX?x?fY?y?.
6. 解：?1?关于X的边缘密度函数为
fX?x???????2x???01dy?2x,0?x?1 f?x,y?dy????0, 其他关于Y的边缘密度函数为
fY?y???????y?11dx?1?,0?y?2??y2 f?x,y?dx??2?0, 其他?1111??22f?x,y?dxdy ?2?P?X?,Y?????????22????120113dy?1dx??2(1?y)dy?.
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