概率论与数理统计(茆诗松)第二版课后第三章习题(2)

（1）(X1, X2, X3)的联合分布列； （2）(X1, X2)的联合分布列． 解：（1）P{(X1,X2,X3)=(0,0,0)}=

8762887570

=，P{(X1,X2,X3)=(0,0,1)}= =， 1312111431312114298577058770

P{(X1,X2,X3)=(0,1,0)}= =，P{(X1,X2,X3)=(1,0,0)}= =，

1312114291312114298544058440

P{(X1,X2,X3)=(0,1,1)}= =，P{(X1,X2,X3)=(1,0,1)}= =，

131211429131211429548405435

P{(X1,X2,X3)=(1,1,0)}= =，P{(X1,X2,X3)=(1,1,1)}= =；

13121142913121114387148510

（2）P{(X1,X2)=(0,0)}= =，P{(X1,X2)=(0,1)}= =，

1312391312395810545

P{(X1,X2)=(1,0)}= =，P{(X1,X2)=(1,1)}= =．

131239131239

X2

01

X1

110/395/39

4． 设随机变量Xi , i =1, 2的分布列如下，且满足P{X1X2 = 0} = 1，试求P{X1 = X2}．

XiP 101

0.250.50.25

解：因P{X1 X2 = 0} = 1，有P{X1 X2 ≠ 0} = 0，

即P{X1 = 1, X2 = 1} = P{X1 = 1, X2 = 1} = P{X1 = 1, X2 = 1} = P{X1 = 1, X2 = 1} = 0，分布列为

故P{X1 = X2} = P{X1 = 1, X2 = 1} + P{X1 = 0, X2 = 0} + P{X1 = 1, X2 = 1} = 0． 5． 设随机变量 (X, Y ) 的联合密度函数为

k(6 x y),0<x<2,2<y<4,

p(x,y)=

0,.其他

试求

（1）常数k；

（2）P{X < 1, Y < 3}； （3）P{X < 1.5}； （4）P{X + Y ≤ 4}． 解：（1）由正则性：∫

+∞+∞ ∞ ∞

∫

p(x,y)dxdy=1，得

2

dx k 6yxy

∫

2

dx∫k(6 x y)dy=∫

2

4