概率论和数理统计 - 复旦大学 - 课后题答案韩旭里 - 写永钦

?5e?5y,fY(y)???0,y?0,其他.

所以

f(x,yX)Y,独立fXx(?f)Yy( )? ??1?0.2?5e?5y???25e?5y,0?x?0.2且y?0,

??0,?0,其他.(2) P(Y?X)???f(x,y)dxdy如图??25e?5ydxdy

y?xD

??0.20dx?x-5y025edy??0.2?0(?5e?x55)dx

=e-1?0.3679.7.设二维随机变量(X,Y)的联合分布函数为

?(1?e?4x)(1?e?2yF(x,y)=?),x?0,y?0,?0,其他.求(X,Y)的联合分布密度. 2【解】f(x,y)??F(x,y)?8e?(4x?2y)?x?y??,x?0,y?0,?0,其他.

8.设二维随机变量(X,Y)的概率密度为

f(x,y)=?4.8y(2?x),0?x?1,0?y?x,??0,其他.求边缘概率密度. 【解】f??X(x)????f(x,y)dy

?x =???04.8y(?2xy)?d??2.4x2(?2x),?0x?,??0,?0,其他.

1 fY(y)??????f(x,y)d x?1 =?4.8y(?2xx)2??y?d??2.4y(?3y4?y),?y0???0,?0,其他.

45

1,

题8图 题9图

9.设二维随机变量(X,Y)的概率密度为

??e?yf(x,y)=,0?x?y,?0,其他.

求边缘概率密度. 【解】fX(x)????(??fx,y)dy

??? =??xe?ydy???e?x ?,x?0,

??0,?0,其他.fY(y)??????f(x,y)dx

?y =???0e?ydx???ye?x,y?0,

??0,?0,其他.

题10图

10.设二维随机变量(X,Y)的概率密度为

f(x,y)=??cx2y,x2?y?1,?0,其他.

(1) 试确定常数c; (2) 求边缘概率密度. 【解】(1)

?????,?????f(xy)dxdy如图??f(x,y)dxdy

D =?1dx?1cx2ydy?4-1x221c?1.

得?c?214.

(2) fX(x)??????f(x,y)dy

46

?1 ????212?21x24xydy???8x2(1?x4),?1?x?1,

??0,??0,其他.fY(y)??????f(x,y)dx

??y212????y4dx???75xy?2y2,0?y?1,

??0,??0, 其他.11.设随机变量(X,Y)的概率密度为

f(x,y)=??1,y?x,0?x?1,?0,其他.

求条件概率密度fY|X(y|x),fX|Y(x|y).

题11图

【解】fX(x)??????f(x,y)dy

?x ?????x1dy?2x,0?x?

1??0,其他.?1???y1dx?1?y,?1?y?0,f??Y(y)????f(x,y)dx???1?0?y?1,??y1dx?1?y,?0,其他.??所以

f(y|x)?f(x,y)??1,|y?|x?1,Y|Xf(x)??

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