习题4
1．设离散型随机变量X具有概率分布律(
X
(2
(1
0
1
2
3
pk
0(1
0(2
0(2
0(3
0(1
0(1
试求E(X)( E(X2(5)( E(|X|)(
解 E(X)(((2)(0.1(((1)(0.2(0(0.2(1(0.3(2(0.1(3(0.1
(0.4(
E(X2(5)(E(X2)(5
(((2)2(0.1(((1)2(0.2(02(0.2
(12(0.3(22(0.1(32(0.1
(2.2(5(7.2(
E(|X|)(|(2|(0.1(|(1|(0.2(|0|(0.2(|1|(0.3(|2|(0.1(|3|(0.1
(1.2.
5．设随机变量X具有概率密度
(
(1)求常数A(
解 由( 得(
(2)求X的数学期望(
解 (
6．设随机变量X的概率密度为
(
求E(3X)( E((2X(5)( E(e(3X)(
解 因为( 所以
E(3X)(3E(X)(3(2(6(
E((2X(5)((2E(X)(5((2(2(5(1.
.
17．(1)求第1题中X的方差D(X)(
解 E(X2)(((2)2(0.1(((1)2(0.2(02(0.2(12(0.3(22(0.1(32(0.1
(2.2(
D(X)(E(X2)([E(X)]2(2.04.
(2)求第14题中X的方差D(X)(
解 已知E(X)=0.1(
E(X2)(((2)2(0.3(02(0.35(22(0.35(2.6(
D(X)(E(X2)([E(X)]2(2.59.
21．设随机变量(X( Y)具有联合概率密度
(
试求(1)X的边缘密度( (2) Y的边缘密度( (3)E(X)( D(X)( (4)E(Y)( D(Y)( (5)X与Y是否不相关？(6)X与Y是否相互独立？
解 (
(1)当|x|(1时( f(x( y)(0( 所以fX(x)(0(
当(1(x(1时( (
所以(
(2)同理得(
(3)(
(
(4)由对称性知E(Y)(0( .
(5) (
所以cov(X( Y)(0( X和Y不相关.
(6)因为f(x( y)(fX(x)(fY(y)( 所以X与Y不相互独立(
24．设已知三个随机变量X( Y( Z中( E(X)(1( E(Y)(2( E(Z)(3( D(X)(9( D(Y)(4( D(Z)(1( ( ( (
(1)求E(X(Y(Z)(
(2)D(X(Y(Z)(
(3)D(X(2Y(3Z)(
解 (1)E(X(Y(Z)(E(X)(E(Y)(E(Z)(1(2(3(6.
(2)

.
(3)D(X(2Y(3Z)(D(X)(4D(Y)(9D(Z)

.
26．设某公路段过往车辆发生交通事故的概率为0(0001( 车辆间发生交通事故与否相互独立( 若在某个时间区间内恰有10万辆车辆通过( 试求在该时间内发生交通事故的次数不多于15次的概率的近似值(
解 设在某时间内发生交通事故的次数为X,则
X~B(100000,0.0001)(
由二项分布的性质知
E(X)(10,D(X)(9.999(
由中心极限定理知
(
28．设某学校有1000名学生( 在某一时间区间内每个学生去某阅览室自修的概率是0(05( 且设每个学生去阅览室自修与否相互独立( 试问该阅览室至少应设多少座位才能以不低于0(95的概率保证每个来阅览室自修的学生均有座位？
解 设至少应设a张座位才能以不低于0.95的概率保证来阅览室的学生都有座位( 并设在同一时间内去阅览室的学生人数为X( 则由题意知
X~B(1000,0.05)( E(X)=50( D(X)=47.5(
由中心极限定理知
(
查表得 (
所以a(61.4( 即至少应设62张座位才能达到要求(