内容发布更新时间 : 2025/5/2 23:53:20星期一 下面是文章的全部内容请认真阅读。
?【解】(1)由??xlim???F(x)?1?A?1??xlim?0?F(x)?得?
xlim?0?F(x)?B??1(2) P(X?2)?F(2)?1?e?2?
P(X?3)?1?F(3)?1?(1?e?3?)?e?3?
(3) f(x)?F?(x)????e??x,x?0?0,x?0
8.设随机变量X的概率密度为
?x,0?x?1,f(x)=??2?x,1?x?2, ??0,其他.求X的分布函数F(x),并画出f(x)及F(x). 【解】当x<0时F(x)=0
当0≤x<1时F(x)??xx??f(t)dt??0??f(t)dt??0f(t)dt ??x0tdt?x2 2
当1≤x<2时F(x)??x??f(t)dt
??01??f(t)dt??f(t)dt??x01f(t)dt??1x0tdt??1(2?t)dt
1x2?
2?2x?32?2?x2?2?2x?1当x≥2时F(x)??x??f(t)dt?1
??0,x?0?x20?x?1故 F(x)???2,?2??x?2x?1?x?2?21,?1,x?2
9.设随机变量X的密度函数为
9
(1) f(x)=ae-?|x|,λ>0;
?(2) f(x)=?bx,0?x?1,?11?x?2, ?x2,?0,其他.试确定常数a,b,并求其分布函数F(x). 【解】(1) 由
????|x|?2a??f(x)dx?1知1?????aedx?2a?0e??xdx??
故 a??2
??e??x,x即密度函数为 f(x)????2?0??e?x??2x?0当x≤0时F(x)??xx???f(x)dx??e?xdx?12e?x??2 当x>0时F(x)??x0??x?x??f(x)dx????2edx??x?02e?dx
?1?12e??x 故其分布函数
?1?1e??x,x?F(x)????20
?1??2e?x,x?0(2) 由1???21??f(x)dx??10bxdx??x2dx?b112?2 得 b=1
即X的密度函数为
??x,0?x?1f(x)???12,1?x?2
?x??0,其他当x≤0时F(x)=0 当0
??xdx?x20x2
当1≤x<2时F(x)??x??f(x)dx??00dx??1x1??0xdx??1x2dx
10
?31? 2x当x≥2时F(x)=1 故其分布函数为
?0,?2?x,?F(x)??2?3?1,?2x?1,?
x?00?x?1
1?x?2x?210.设随机变量X的分布函数为:F(x)=A+Barctanx,(-??x???). 求:(1)系数A与B;
(2)X落在(-1,1)内的概率; (3)X的分布密度。
1A=1/2,B=解 ○
12
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