???0axedx????x??0axde???x??axe?x|0??????0aedx?x
????????????????????ae?x|0?a得到 a?1.
x?0,F(x)?0, x?0,F(x)??x0tedt???tde0?x?tx?t??te?x?t|0??edt0?xxx?t
?????????????????????xe?e?t|?1?xex0?e,?0,??????????????????????x?0,F(x)??. ?x?x1?xe?e,???x?0.?
11. 解 1) 不能,由于不是单调不减；
2) 不能,由于不是单调不减；
3) 能,其他场合定义F(x)?1,x?0．
?2x,??0?x?1,12. 解 1) 是连续型随机变量，f(x)??
?0,??????其他.2) 不是，由于连续型随机变量取值与一点的概率为0，而P(X?1)?1/2．
13. 解 由规范性得到?????f(x)dx?1，
?x?????ae?|x|dx???12??02ade?2ae?x|0?2a ,
??得到 a?.
x?0,F(x)?x?0,F(x)???x??0??1212edt?exdx??t12x0e?x,
?t?12edt?12?e?t|0?1?x12e?x,
?1?xe,??????????????????????x?0,??2. F(x)???1?1e?x,?????????????????x?0.??2
214. 解 由方程 4x?4?x???2?0有实根得到
??16??16(??2)?0，
2解得 ??2????or????????1,
由于????U(0,5)，所以P(??2)?3/5．
15. 解 设Y为四次取值大于发生a的次数，则Y?b(4,p)，其中
p?P(X?a)?1?a,(0?a?1)
又 P(Y?1)?1?P(Y?0)?0.9，P(Y?0)?a?0.1 解得 a?0.5623.
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16. 解 1) P(X?a)?P(??1X?52?a?52a2)?0.9,查标准正态分布表得到
(0.9)?1.2816,解得a? 7.5631.
|X?5|2?)?0.01,
2) P(|X?5|?a)?P(P(|X?5|2?aaa)?2??()?0.01,得到?()=0.995 222查标准正态分布表得到??1(0.995)=2.5758,a? 5.1517.
17. 解 设优秀的最低分为a,数学成绩为X,根据条件得到
P(X?a)?0.05,P(X?7010?a?7010)?0.95
查标准正态分布表得到??1(0.95)?1.645,解得 a?86.45.
18. 解
R 10 11 12 13 ξ 20π 22π 24π 26π η 100π 121π 144π 169π P 0.1 0.4 0.3 0.2
19. 证明 当a????y?b???时
F(y)?P(Y?y)?P(?X???y)?P(X?y??y???)
?p(y)???a1b?adx?1b?a(y????a)?y???a?b??a?,
1b??a?,a????y?b???,
其他 p(y)?0
1?,???????????a????y?b???,? p(y)??b??a??0,????????????????????????????????????????????????其他.?即Y服从[a???,b???]的均匀分布,
20. 解 1) 当y?0时
1F(y)?P(Y?y)?P(X?y)?P(X?y3)
13 ?p(y)??y301edx?1?e?213?x?y3,
13其他 p(y)?0
?21ye?y3,y?0,
1?13?y3??y?ye,?????y?0,?1?e3,??????????y?0,p(y)??3,F(y)??
??0,????????????????????其他.?0,????????????????????其他.? 14
2) 当0?y?1时
F(y)?P(Y?y)?P(e?X?y)?P(X??lny)?????lny?xedx?y,
p(y)?y,0?y?1, 其他 p(y)?0.
,y?0,?0?????????y,??????y?1,??,?y?1 F(y)??y???????? ,p(y)??0,?????????其他.??1?????????y?1?,
21. 解 1) 当1?y?e时，
F(y)?P(Y?y)?P(eX?y)?P(X?lny)??lny0dx,
p(y)?1y,1?y?e,
其他 p(y)?0
?1?,??????????????????1?y?e, p(y)??y?0,??????????????????其他.?2) 当0?y???时
?yF(y)?P(Y?y)?P(2lnX?y)?P(X?ep(y)?1?y2)??1?ye2dx.,
2其他 p(y)?0.
e2,0?y???,
?1?2y?e,????????????0?y???, p(y)??2?0,??????????????????其他.?3) 当1?y???时 F(y)?P(Y?y)?P(p(y)?1y21X?y)?P(X?1y)??11ydx.,
,1?y???,
其他 p(y)?0
?1?2,????????1?y???, p(y)??y?0,??????????????????其他.?
22. 解 1) 当y?0时
F(y)?P(Y?y)?P(eX?y)?P(X?lny)??lny??12?e?x22dx.,
p(y)?1y12?e?lny22,y?0,
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其他 p(y)?0
lny?11?e2,????y?0,? p(y)??y2??.?0,??????????????????????其他.22) 当y?0时
F(y)?P(Y?y)?P(e?X?y)?P(X??lny)?????lny12?e?x22dx.,
p(y)?1y12?e?lny22,y?0,
其他 p(y)?0
lny?11?e2,????y?0,? p(y)??y2??.?0,??????????????????????其他.3) 当y?0时
2F(y)?P(Y?y)?P(|X|?y)?P(?y?X?y)??y?y12?e?x22dx.,
p(y)?212? 其他 p(y)?0
e?y22,y?0,
y??1e2,???????y?0,?2p(y)?? 2??.?0,??????????????????????其他.2
23. 证明 不妨设a?0
F(y)?P(Y?y)?P(aX?b?y)?P(X?y?b1), aa同理可得a?0的情形. f(y)?fX(y?bay?b)??a??fX(x)dx
24. 解 由上题的结论可得
y??时,f(y)?y??,f(y)?0
??e??y???
分布密度函数为
??????y?,?????y??,?e f(y)????0,????????????????y??.?
25. 解 设年化收益率为r,r?lnXX0?lnX?lnX0
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