?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.
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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
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?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)??