1-32 已知三维高斯随机变量(X1,X2,X3)各分量相互独立，皆服从标准高斯分布。求Y1?X1?X2和Y2?X1?X3的联合特征函数？
?????0??100M??0???X???C?X???010?
?0???001??
思路：Y??是?X??线性变换故也服从高斯分布，求得可以写出联合特征函数
??Y?X1??X1?1?X1?X2?Y???Y1?????110???2?X1?X3?Y2??101??X??2?AX????X3??2? ???X3?Y??AX?，线性变换，故Y??也服从高斯分布
??M??AM?????0YX??CACT??21??0??Y?XA??12??
N维高斯变量的联合特征函数
QjU???TY????????T??UTC??Y??1,?,?n??E??YU???e???exp??jMYU???2?????22??
?exp??1??1?2??2?
??M?YCY就
2、已知随机变量(X,Y)的联合概率密度为
?6xy(2?x?y)0?x?10?y?1 fXY(x,y)??0else?
(1)条件概率密度f(xy),f(yx)
(2)X和Y是否独立？给出理由。
解题思路：f(x,y)?fX(x),fY(y)?f(xy),f(yx)
解：(1)
fX(x)?????12???06xy(2?x?y)dy?4x?3x0?x?1fXY(x,y)dy???else?0?6y?2?x?y?fXY(x,y)?0?x?10?y?1fY(yx)???4?3xfX(x)?0else??6x?2?x?y?0?x?10?y?1?同理fX(xy)??4?3y?0else?
(2) fX(xy)?fX(x)orX和Y不相互独立
fXY?x,y??fX?x?fY?y?
4、已知 (X1,X2,X3) 是三维高斯变量，其期望和方差为
?X1???X??X?2??m1??0?????0?MX??m?2????732??CX??341?? Y1?X1?X2 Y2?X3
??X3????m3????0????212??求：(1) (X1,X2)的边缘特征函数。
(2) (Y1,Y2)的联合概率密度
高斯变量的线性变换后仍服从高斯分布
所以(X?1,X2)、Y服从高斯分布
(1) E??X1??X????0??C?73?X1X2???2??0??34? ?22Qexp??7u1?6u1u2?4u2?X?u1,u2???2??
(2) A???110???0??17?001??MY???0??CY???3C?1?1?2?3?Y?25CY25???317??
3?2?? 22??2Y?6Y?Y?17Y?1112?2fY?y1,y2??exp???
10?50??
