Probability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION
Roy D. Yates David J. Goodman
08_1 Yates Chap. 7 1
Probability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION
Roy D. Yates David J. Goodman
Definitions, Theorems, Proofs, Examples, Quizzes, Problems, Solutions
Chapter 7
08_1 Yates Chap. 7 2
08_1 Yates Chap. 7 3
Theorem 7.1
[ ] [ ] [ ] [ ]
The sample mean ( ) has expected value and variance
( ) , ( ) .
n
n n
M X
Var X
E M X E X Var M X
= = n
08_1 Yates Chap. 7 4
[ ] [ ] [ ] [ ]
The sample mean ( ) has expected value and variance
( ) , ( ) .
n
n n
M X
Var X
E M X E X Var M X
= = n
08_1 Yates Chap. 7 5
Section 7.2
Deviation of a Random Variable from the Expected Value
08_1 Yates Chap. 7 6
08_1 Yates Chap. 7 7
08_1 Yates Chap. 7 8
Revisit
08_1 Yates Chap. 6 9
08_1 Yates Chap. 6 10
08_1 Yates Chap. 7 11
08_1 Yates Chap. 7 12
08_1 Yates Chap. 7 13
08_1 Yates Chap. 7 14
08_1 Yates Chap. 7 15
08_1 Yates Chap. 7 16
08_1 Yates Chap. 7 17
08_1 Yates Chap. 7 18
08_1 Yates Chap. 7 19
08_1 Yates Chap. 7 20
08_1 Yates Chap. 7 21
08_1 Yates Chap. 7 22
08_1 Yates Chap. 7 23
08_1 Yates Chap. 7 24
08_1 Yates Chap. 7 25
08_1 Yates Chap. 7 26
08_1 Yates Chap. 7 27
08_1 Yates Chap. 7 28
[ ]
If has finite variance, then the sample mean ( ) is a sequence of consistent estimates of .
X M
nX
E X
Theorem 7.7
08_1 Yates Chap. 7 29
[ ]
If has finite variance, then the sample mean ( ) is a sequence of consistent estimates of .
X M
nX
E X
08_1 Yates Chap. 7 30
08_1 Yates Chap. 7 31
08_1 Yates Chap. 7 32
08_1 Yates Chap. 7 33
08_1 Yates Chap. 7 34
08_1 Yates Chap. 7 35
Theorem 7.10
( ) 1 [ ] .
n
E V X n Var X
n é ù = -
ë û
08_1 Yates Chap. 7 36
( ) 1 [ ] .
n
E V X n Var X
n é ù = -
ë û
08_1 Yates Chap. 7 37
08_1 Yates Chap. 7 38
08_1 Yates Chap. 7 39
08_1 Yates Chap. 7 40
08_1 Yates Chap. 7 41
