(B)
1.随机变量X与Y均服从正态分布。X～N(μ,42),Y～N(μ,52),证P1=P?X???4?,P2=P?Y???5?,则() (A)对任何实数μ，都有P1?P2 (B)对任何实数μ，都有P1?P2 (C)只对μ的个别值，才有P1=P2 (D)对任何实数μ，都有P1>P2
?X???解：p1?P?X???4??P???1?????1??4??Y????Y???p2?P?Y???5??P??1?1?P??1??1???1?
?5??5?????X??1???X?????1??1???1?即P1?P2故（A）成立。
2.设随机变量X～N(μ,?2)，则随着?的增大，概率P?X?????()
(A)单调增大(B)单调减小(C)保持不变(D)增减不定 解：?X～N(μ,?2)
?X???～N(0,1)
?X??? ?P?X??????P??1????1?????1??2??1??1?故（C）成立。???
3.设F1(x)与F2(x)分别为随机变量X1与X2的分布函数,为使F(x)=aF1(x)-bF2(x)是某随机变量的分布函数,在下列给定的各组数值中应取() (A)a=35,b=-25
(B)a=23,b=23 (C)a=-12,b=32 (D)a=12,b=-32
解：?F1?x?,F2?x?都是分布函数，?F1?????1,F2?????1为使F?x??aF函数，必满足条件：F?????1 1?x??bF2?x?是某一随机变量的分布即：a?b?1故(A)成立 (C)计算题
1.设测量误差X～N(0,102),试求在100次独立重复测量中,至少有三次测量误差的绝对值大于19.6的概率α,并用泊松分布求出α的近似值(要求小数点后取两位有效数字). λ e??1 2 3 4 5 6 7 0.368 0.135 0.050 0.018 0.007 0.002 0.001
解：设100次独立重复测量中有Y次测量误差的绝对值大于19.6，则Y～B（100，p）,p=P?X?X～N(0,1) 10?19.6?
?X??X??p?P?X?19.6??P??1.96??1?P??1.96??2?1???1.96???2(1?0.975)?0.05?10??10?012??P?Y?3??1??P?Y?i??1?(C100?0.050?0.95100?C100?0.051?0.9599?C100?0.052?0.9598)i?02100?99?0.052?0.9598)?0.8817370192若用泊松分布求?的近似值，因??100?0.05?5则有?1?(0.95100?5?0.9599?50?551?552?5??1?e?e?e?1?18.5e?5?1?18.5?0.007?0.87
0!1!2!
2.一实习生用同一台机器接连独立地加工3个同种零件,第i个零是不合格品的概率Pi?1,(i=1,2,3,),此 X 表示1?i3个
零件中合格品的个数,求P{X=2}.
解：令Ai?“加工的第i个零件是合格品”（i?1,2,3）由题意A1,A2,A3相互独立,且P(Ai)?1??“X?2”?A1A2A3?A1A2A3?A1A2A312111312311?P?X?2??P(A1A2A3)?P(A1A2A3)?P(A1A2A3)??????????234234234241i?(i?1,2,3)1?i1?i
3.设随机变量X的绝对值不大于1,P{X=-1}=1,P{X=1}=1.在
84事件{-1 率与该子区间的长度成正比,求: (1)X的分布函数F(x)=P{X≤x};(2)X取负值的概率 115解:P?X?1??0,P?X?1??1,P?X?1??1?P?X?1??1???848对任意的(c,d)?(?1,1)有PX?(c,d)X?1?k(d?c),其中k为比例系数，??特别地，取（c,d）?(?1,1)得1?PX?(?1,1)x?1?k?1???1???2k?k?12(1)当x??1时，F(x)?0,当x?1时，F(x)?1.故可设定?1?x?1，由全概率公式??P?X?x??P?X?x,X?1??P?X?x,X?1??P?X?x,X??1??P?X?x,X?1?5x???1?1?P?X?x,X?1??P?X?x,X??1??P?X?1?P()X?xX?1?P?X??1????828?0x??1?1故F(x)???5?x?1??2?,?1?x?1?16x?11?7（2）?F(x)在(?1,1)内是连续的，?P?X?0??F(0?0)?F(0)?16??