11于是，E?XY????xydxdy??dx?xydy
2G20新疆财经大学数学考研辅导班教学资料0
1??xy24022?x022?x111?14162?dx??x?x?2?dx??x3?4x2?4xdx??x4?x3?2x2??.
40404?433?022??2111同理，E?X????xdxdy??dx?xdy??xy2G2020022?x22?x01dx??x?2?x?dx
2021?1?2??x2?x3??； 2?3?0311E?Y????ydxdy??dx2G2022?x212ydy?y??40022?x011?12?2dx???x?2?dx??x3?2x2?4x??.
404?3?0322于是，cov?X,Y??E?XY??E?X?E?Y??而EX162216444?????. 33339922??22?x1111???x2dxdy??dx?x2dy??x2ydx??x2?2?x?dx
02G202020022?x1?21?2??x3?x4??； 2?34?03EY2??211???y2dxdy??dx2G20222?x13ydy?y??600222?x013dx????2?x?d(2?x)
60214???2?x?24?02； 32且D?X??EX2??2?2?22?2?2?E2?X??????，D?Y??EY2?E2?Y??????，
3?3?33?3?3??2故，?XY?44cov?X,Y?22. ?9?23D?x?D?y?35．设随机变量(X,Y)具有密度函数f(x,y)??试求:E(X)，E(Y)，cov(X,Y). 解：E?X????????1x?1,?0,y?x,0?x?1,其他,，
1?????2xf(x,y)dxdy?dxxdy?2x????dx?0?x02； 3yE?Y??1x??????yf(x,y)dxdy??dx?ydy?0；
0?x?1,1?y?x由区域的对称性及函数的相对奇偶性可知：
x?1E?XY???dx?xydy?0；
0?x1xoy??xx?1,?1?21 故，cov(X,Y)?E?XY??E?X?E?Y??0.
7．设随机变量X,Y,Z满足: E(X)?E(Y)?1，E(Z)??1，D(X)?D(Y)?D(Z)?1，
新疆财经大学数学考研辅导班教学资料
?XY?0，
?1XZ???YZ?2，试求：E(X?Y?Z)，D(X?Y?Z). 解：E(X?Y?Z)?E?X??E?Y??E?Z??1?1?1?1；
D(X?Y?Z)?D?X?Y??D?Z??2cov?X?Y,Z?
?D?X??D?Y??2cov?X,Y??D?Z??2cov?X,Z??2cov?Y,Z?
?D?X??D?Y??2D?X?D?Y???X,Y??D?Z??2D?X?D?Z???X,Z? ?2D?Y?D?Z???Y,Z?
?1?1?2?1?1?0?1?2?1?1?12?2?1?1???1???2???3.
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