第三章****题 3
1 、设 A,B 是相互独立且同服从 N (0,s 2 ) 的随机变量,求随机过程的
X t = A t + B, tÎ }均值函数、自相关函数、协方差函数。答:
均值函数: mX t = E (X t ) = E (At + B ) = tE (A ) + E (B) = 0
自相关函数:
t2 )
ë (
)(
)û
R
( 1
2 )
= E
(
X
t1
1
+ B
At
2
t , t
X
= E é
At
+ B ù
)
( 1
2 )
(
)
ë 1 2
A2 +
( 1
+ t
2 )
û
1 2
(
)
+ E
(
+
+ t
= E é t t
t
AB + B 2 ù
= t t
E
A 2
B 2
t
E
AB
A, B 相互独立,\ E (AB ) = E (A ) E (B) = 0
E (A2 ) = E (B2 ) = s 2 ,\ R ( t1 , t 2 ) = ( t1t 2 +1)s 2
协方差函数:
c X (t1 , t 2 ) = R ( t1 ,t 2 )- E (X t1 )E (Xt2 )
E (Xt ) = 0 ,\ c X (t1 , t 2 ) = R ( t1 , t 2 ) = ( t1t 2 +1)s 2
2、设随机过程{X t , t ÎT}的均值函数为 mxt ,协方差函数为 c X (t1 , t2 ) 。记随机过
Yt = X t + j (t ), t ÎT ,其中,j (t )是普通函数。(1)求Yt 的均值函数和协方差函数;
(2)如果j ( t ) = -mXt
,证明 RY (t1 , t 2 ) = cY (t1 , t 2 ) = c X (t1 ,t2 )
答:
(1)
mYt
= E (X t + j ( t )) = E (X t )+ j ( t ) = m X t
+j (t )
RY
(t1 , t 2
) = E é ( X t
+ j ( t1 ))(X t
+ j ( t 2 ))ù
= E é X t
X t
+ X t j ( t 2
)+ X t
j ( t1 )+j ( t1 )j (t2
)ù
ë
1
û
ë1
1
û
RX (t1 , t 2 )+ j ( t 2 )m X t1 + j ( t1 )m X t 2 +j ( t1 )j (t2 )
cY (t1 , t 2 ) = RY (t1 , t 2
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