FZ(z)?P?Z?z??P?X?Y?z???dx?????z?x??f(x,y)dy?y?u?x?????dx?z??f(x,u?x)d?u?x? ??dx?????z?????f(x,u?x)du???f(x,u?x)dx?du????????z
因此，fZ(z)??????f(x,z?x)dx.又
????fZ(z)??????f(x,z?x)dx??f(z?y,y)d?z?y???????????f(z?y,y)dy
因此，fZ(z)??????f(x,z?x)dx??f(z?y,y)dy.
当X与Y独立时，f(x,z?x)?fX(x)fY(z?x)，f(z?y,y)?fX(z?y)fY(y)，这样，
fZ(z)??????fX(x)fY(z?x)dx??????fX(z?y)fY(y)dy.
2.随机变量差的分布
设?X,Y?为二维连续型随机变量，其联合概率密度函数为f(x,y)，边缘概率密度函数为fX(x)，fY(y).则Z?X?Y的概率密度函数为fZ?z??互独立，则fZ?z??事实上，
?????f(u?z,z)du.若X与Y相
?????fX?u?z?fY(z)du.
??y?zFZ?z??P?Z?z??P?X?Y?z??P?X?Y?z???dy?????f(x,y)dx
?u?x?y?????dy?z??f(u?y,y)d?u?y?????f(u?y,y)du?dy????????z??因此，fZ?z???????f(u?z,z)du.若X与Y独立，则f(u?z,z)?fX?u?z?fY?z?.因此，
fZ?z???????fX?u?z?fY(z)du.
3.随机变量积的分布
设?X,Y?为二维连续型随机变量，其联合概率密度函数为f(x,y)，边缘概率密度函数为fX(x)，fY(y).则Z?XY的概率密度函数为
fZ?z?????????1?z?1?z?f?x,?dx??f?,y?dy.又若X，Y相互独立，则
??yx?x??y? 26
fZ?z???事实上，
??????1??1?z?1?z??z?f?x,?dx??fX?x?fY??dx??fX??fY?y?dy.
??x??yx?x??x??y?FZ?z??P?Z?z??P?XY?z???dx?f(x,y)dy??dx?zf?x,y?dy
0??x??zx??0???
??0dx?zx??f(x,y)dy?u?xy???0dx?z??z???1z???1?u??u??u???u??f?x,?d??????f?x,?dx?du????f?x,?dx?du??0x?x???x??x????0x?x????0??dx?zf?x,y?dy?x??u?xy?0??dx???zz?01z?01?u??u??u???u??f?x,?d??????f?x,?dx?du????f?x,?dx?du?????x????xxxx????????x????这样，
z???1z?01z???1?u???u???u??FZ?z?????f?x,?dx?du????f?x,?dx?du????f?x,?dx???0????x????xx?x???x???x?????于是，fZ?z???????1?z?f?x,?dx. x?x???zy??0??FZ?z??P?Z?z??P?XY?z???dy?f(x,y)dx??dy?zf?x,y?dx
0??y???0dy?zy??f(x,y)dx?u?xy???0dy?0z??z???1z???1?u??u??u???u??f?,y?d??????f?,y?dy?du????f?,y?dy?du??0y?y???y??y????0y?y????0??dy?zf?x,y?dx?y??u?xy???dy???zz?01z?01?u?u?u???u??f?,y?d????f?,y?dy?du????f?,y?dy?du????????y?y?y?y????y?y???这样，
fZ?z???????1?z?f?,y?dy.因此， y?y?z???1z?01z???1?u???u???u??F?z?????f?,y?dy?du????f?,y?dy?du????f?,y?dy?du??0????y????yy?y???y???y???????于是，fZ?z?????1?z?f?,y?dy. y?y?fZ?z?????????1?z?1?z?f?x,?dx?fZ?z???f?,y?dy.又若X，Y相互独立，则
??yx?x??y? 27
?z??z?f?x,??fX?x?fY??，?x??x??z??z?f?,y??fX??fY?y?，这样， ?y??y?fZ?z?????????1?z?1?z?fX?x?fY??dx??fX??fY?y?dy.
??yx?x??y?4.随机变量商的分布
设?X,Y?为二维连续型随机变量，其联合概率密度函数为f(x,y)，边缘概率密度函数为fX(x)，fY(y).则Z?独立的，则fZ?z??事实上，
??X的概率密度函数为fZ?z???yf(zy,y)dy.又若X，Y是
??Y?????yfX?zy?fY?y?dy.
XX?X?????FZ?z??P?Z?z??P??z??P?Y?0,?z??P?Y?0,?z?YY?Y????? ??dy?0??yz??f(x,y)dx??dy???u?xy0??yzf(x,y)dx?????dy?yf(uy,y)du???yf(uy,y)dy?du???????0?zz?.
??0dy?yz??f(x,y)dx?x?uy0???dy?z??f(uy,y)d?uy???0u??0??dy???yzf(x,y)dx?xyx?uy???0dy???zf(uy,y)d?uy???dy???0z0??yf(uy,y)du?yf(uy,y)dy?du??????????????z这样，
??z0z???????FZ?z????yf(uy,y)dydu???yf(uy,y)dydu???yf(uy,y)dy?du?????????????0?????????z于是，fZ?z??????? yf(zy,y)dy.若X，Y相互独立，则f(zy,y)?fX?zy?fY?y?.于是，
??fZ?z???????yf(zy,y)dy????yfX?zy?fY?y?dy.
5.随机变量的最值的分布
假设X1，X2，……，Xn联合分布函数为F?x1,x2,......,xn?.Z?max?Xi?.则
1?i?nFZ?z??P?Z?z??Pmax?Xi??z?P?X1?z,X2?z,......,Xn?z??F?z,z,......,z?1?i?n??若X1，X2，……，Xn是独立的，则FZ?z??FX1(z)FX2(z)......FXn(z).又设Y?min?Xi?，
1?i?n 28
FZ?z??P?Z?z??Pmin?Xi??z?1?Pmin?Xi??z?1?P?X1?z,X2?z,......，Xn?z?1?i?n1?i?n????若X1，X2，……，Xn是独立的，则
FZ?z??1?P?X1?z?P?X2?z?......P?Xn?z??1???1?P?X1?z?????1?P?X2?z???......??1?P?Xn?z??? ?1???1?FX1(z)????1?FX2(z)??.......??1?FXn(z)??
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第四章 随机变量的数字特征
一.期望
1.离散型随机变量期望的定义
设离散型随机变量X的分布律为P(X?xk)?pk，若级数
?xkkpk绝对收敛，则称
E(X)??xkpk为随机变量X的数学期望，简称期望，也叫均值.
k2.连续型随机变量的数学期望的定义
设连续型随机变量X的概率密度函数为f(x)，若广义积分
?????则称 xf(x)dx绝对收敛，
E(X)??????xf(x)dx为随机变量X的数学期望，简称期望，也叫均值.
3.随机变量的函数的数学期望的求法
随机变量的函数的数学期望求法以如下定理为依据：设Y是随机变量X的函数：
Y?g(X)(g是连续函数).
k?1,2,......，(i) 如果X是离散型随机变量，它的分布律为P?X?xk??pk，若
绝对收敛，则有E(Y)?E(g(X))??g(x)pkkk?g(x)pkk?1??k.
??(ii) 如果X是连续型随机变量，它的概率密度为f(x)，若
???f(x)g(x)dx绝对收敛，则有
E(Y)?E(g(X))??????g(x)f(x)dx.
4.多维随机变量的函数的数学期望的求法
多维随机变量的函数的数学期望求法以如下定理为依据： 设?X1,X2,......,Xn?为多维随机变量，g为n元连续函数.
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