X pk 0 1222121 9222 122 E(X)?
4、设X为所得分数 P{X?k}? P{X?k}? E(X)?491216136,,k?1,2,3,4,5 k?7,8,9,10,11,12
5、（1）由P{X?5}?P{X?6}，则
?55!e????66!e??
解出??6，故E(X)???6
? （2）由于?(?1)k?1k?1k6?k22?6??2?k?1(?1)k?11k不是绝对收敛，则E(X)不存在。
6、（1）E(X)? （2）E(X)?7、E(X)?8、E(X)?9、E(X)???????xf(x)dx???5??0x?19xe?x3dx?6
??????xdF(x)???xd(1?25x52)?50?141x25dx?10
?????????xf(x)dx?xf(x)dx?xf(x)dx?kk???102x?42x(1?x)dx?x?2(1?x?321x2
????10)dx?3?2ln2
2???1x(1?x)dx??10x?32x(1?x)dx?0
210、由P{X?k}?C4p(1?p)E(sin4?k,k?0,1,2,3,4
?X2)?4p(1?p)(1?2p)
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11、R的概率密度为
?1?, f(x)??a??0,??0?x?a其它
E(V)?12、
??x63???f(x)dx??a?x630?1adx??24a
3E(g(X))??????g(x)f(x)dx??40x?2310xe?310xdx????416?310xe?310xdx?2009?4409e?65
13、Y1的分布函数为
0,??n(y1)??1?(1?y1),?1,?y1?00?y1?1 y1?1FminY1的概率密度为 fmin?n(1?y1)n?1,(y1)???0,0?y1?1其它
1n?1E(Y1)??????y1fmin(y1)dy1??10y1?n(1?y1)n?1dy1?
Yn的分布函数为
?0,?n(yn)??yn,?1,?yn?00?yn?1
yn?1FmaxYn的概率密度为
fmax?nynn?1,(yn)???0,0?yn?1其它
n?1E(Yn)??????ynfmax(yn)dyn??10yn?nyndyn?nn?1
14、X的分布律为 X
0 1 2 22
pk
1528 1228 128 Y的分布律为 Y pk E(X)?12,0 10281 15282 34328 E(Y)?
E(XY)?1?P{X?1,Y?1}?2?(P{X?1,Y?2}?P{X?2,Y?1})?4?P{X?2,Y?2}?314
E(X?Y)?1?(P{X?1,Y?0}?P{X?2,Y?1})?2?P{X?2,Y?0}?(?1)?(P{X?0,Y?1}?P{X?1,Y?2})?(?2)?P{X?0,Y?2}??1 4E(3X?2Y)?2?P{X?0,Y?1}?4?P{X?0,Y?2}?3?P{X?1,Y?0}?5?P{X?1,Y?1}?7?P{X?1,Y?2}?6?P{X?2,Y?0}?8?P{X?2,Y?1}?10?P{X?2,Y?2}?3
15、
E(min(X,Y))?1?(P{X?1,Y?1}?P{X?1,Y?2}?P{X?2,Y?1})?2?P{X?2,Y?2}?314
E(Y/(X?1))?1?(P{X?0,Y?1}?P{X?1,Y?2})?2?P{X?0,Y?2}?12?P{X?1,Y?1}?11?x13P{X?2,Y?1}?225?P{X?2,Y?2}?9 1416、E(X)?
E(Y)???00dx?0x?24xydy?y?24xydy?1dx?1?x525
2150E(XY)??10dx?1?x0xy?24xydy?
17、
23
E(Y)?2000?P{X?10}?1000?P{X?11}?(?1000)?P{X?13}?(?2000)?P{X?14}?400
E(Y)?200022?P{X?10}?100022?P{X?11}?(?1000)?P{X?13}62?(?2000)?P{X?14}?1.6?10D(Y)?E(Y)?(E(Y))??22
?1.44?10
x22618、E(X)??0x?x??2e22?dx??22?
E(X)?2???0x?2x?2x?e2?2dx?2?
D(X)?E(X)?(E(X))22?(2??2)?,2D(X)?2??2?
?19、E(X)??k?1k(1?p)k?1p?1p
? E(X)?2?k?1k(1?p)2k?1p?2?pp2
D(X)?E(X)?(E(X))22?1?pp2
20、（1）E(X)???????x?k?xkk?1dx?kk?1?
E(X)不存在。
（2）由于??x??x2dx???，则当k?1时,（3）E(X)?2????x?2k?xkk?1dx?kk?2??
k?222 D(X)?E(X)?(E(X))22(k?1)(k?2)
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（4）由于????x?22?x12,32dx???，则当k?2时,34314956D(X)不存在。
21、（1）E(X)?
E(Y)?,E(XY)?
Cov(X,Y)?E(XY)?E(X)E(Y)??E(X)?21628,2E(Y)?227282
?928,D(Y)?E(Y)?(E(Y))22D(X)?E(X)?(E(X))?45112
?XY?Cov(X,Y)D(X)25,D(Y)??2555,
215275 （2）E(X)?
E(Y)?E(XY)?
Cov(X,Y)?E(XY)?E(X)E(Y)??2E(X)???010dx?1?x0x?24xydy?y?24xydy?2221515,
D(Y)?E(Y)?(E(Y))22 E(Y)?21dx?21?x0D(X)?E(X)?(E(X))?125?125
?XY?Cov(X,Y)D(X)D(Y)??23
275D(X?Y)?D(X)?D(Y)?2Cov(X,Y)?
（3）X的分布律为
X 0 1 2 0.38 2 0.5 pk 0.24 0.38 Y的分布律为 Y pk 0 0.16 1 0.34 E(X)?1.14,E(Y)?1.34
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