?P{X1?0}P{X2?0}P{X3?0}?0.9?0.8?0.7?0.504,
P{X?1}?P{X1?X2?X3?1} ?P{X1?1,X2?0,X3?0} ?P{X1?0,X2?1,X3?0}?P{X1?0,X2?0,X3?1} ?P{X1?1}P{X2?0}P{X3?0} ?P{X1?0}P{X2?1}P{X3?0}?P{X1?0}P{X2?0}P{X3?1} ?0.1?0.8?0.7?0.9?0.2?0.7?0.9?0.8?0.3?0.398, P{X?3}?P{X1?X2?X3?3}?P{X1?1,X2?1,X3?1}
?P{X1?1}P{X2?1}P{X3?1}?0.1?0.2?0.3?0.006.
由P{X?0}?P{X?1}?P{X?2}?P{X?3}?1得出
P{X?2}?1?P{X?0}?P{X?1}?P{X?3} ?1?0.504?0.398?0.006?0.092.因此X的概率分布为
X p 0 0.504 1 0.398 2 0.092 3 0.006 (2)令p1?P{X1?1}?0.1,p2?P{X2?1}?0.2,p3?P{X3?1}?0.3,因Xi均服从0?1分布,故EXi?pi,DXi?pi(1?pi)所以E(X1)?0.1??E(X2)?0.2??E(X3)?0.3,
D(X1)?0.1?0.9?0.09,D(X2)?0.2?0.8?0.16,D(X3)?0.3?0.7?0.21
X?X1?X2?X3.因Xi服从0?1分布, 且X1,X2,X3相互独立,故由数学期望与方差的
性质 EX?E(X1?X2?X3)?EX1?EX2?EX3?0.6.
DX?D(X1?X2?X3)?DX1?DX2?DX3?0.46.
注：X的期望与方差也可以直接用期望与方差的公式来计算：
E(X)?0?P{X?0}?1?P{X?1}?2?P{X?2}?3?P{X?3}?0?0.504?1?0.398?2?0.092?3?0.006?0.6,
D(X)?02?P{X?0}?12?P{X?1}?22?P{X?2}?32?P{X?3}?0?0.504?1?0.398?2?0.092?3?0.006?0.46.
十四、(本题满分4分)
【解析】(1)已知联合概率密度可以直接利用求边缘密度的公式fX(x)?2222
?????f(x,y)dy求出
边缘概率密度.
当x?0时,fX(x)?当x?0时,fX(x)?因此X的密度为
????????0dy?0;
f(x,y)dy??0dy??e?ydy??e?y??xx????x???e?x.
?e?x,x?0,fX(x)??
0,x?0.?(2) 概率P{X?Y?1}实际上是计算一个二重积分,根据概率的计算公式：
P{X?Y?1}?x?y?1??f(x,y)dxdy,
yy?x再由二重积分的计算,化为累计积分求得概率
1 x?y?1 P{X?Y?1}. P{X?Y?1}?x?y?1120??f(x,y)dxdy??dx??(1?x)?x120120O 1?xx12 1 x
e?ydy120?x?12???[e
?e]dx???edx??edx?1?2ex?1?e?1.