Probability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers SECOND EDITION
Roy D. Yates David J. Goodman
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 10
1 Chap. 10
2 Chap. 10
3 Chap. 10
4 Chap. 10
(1) Ensemble Average
(2) Time Average 5
Chap. 10
6 Chap. 10
7 Chap. 10
8 Chap. 10
9 Chap. 10
10 Chap. 10
11 Chap. 10
12 Chap. 10
13 Chap. 10
14 Chap. 10
15 Chap. 10
16 Chap. 10
17 Chap. 10
18 Chap. 10
19 Chap. 10
20 Chap. 10
21 Chap. 10
22 Chap. 10
23 Chap. 10
24 Chap. 10
25 Chap. 10
26 Chap. 10
27 Chap. 10
28 Chap. 10
29 Chap. 10
30 Chap. 10
31 Chap. 10
32 Chap. 10
33 Chap. 10
34 Chap. 10
35 Chap. 10
36 Chap. 10
37 Chap. 10
[ ] [ ]
[ ] (
1) (
1)
( ) ( ) ( )
( ) 1
0 (*)
1 0,
( ) 0 otherwise.
( ) 0,
0 otherwise.
n
n
n
X X X
X n n
x
n n n
x X
x X
dF x dF x
f x
dx dx
F x P X x P X x
P X x P N t x N t e
e x
F x
e x
f x
l
l
l l
-
- -
-
-
= =
= £ = - >
é ù
> = ë + - = û =
ì - ³
= í î
ì ³
= í î
Proof of Theorem 10.3
[ ] ( )
(*) Divide by intervals. In each interval, there is one arrival or none.
sec 1 1
m
m x
m
x m
P no arrivals in x onds p x e
m
l -l
®¥
æ ö
= - = ç - ÷ ®
è ø
[ ] [ ]
[ ] (
1) (
1)
( ) ( ) ( )
( ) 1
0 (*)
1 0,
( ) 0 otherwise.
( ) 0,
0 otherwise.
n
n
n
X X X
X n n
x
n n n
x X
x X
dF x dF x
f x
dx dx
F x P X x P X x
P X x P N t x N t e
e x
F x
e x
f x
l
l
l l
-
- -
-
-
= =
= £ = - >
é ù
> = ë + - = û =
ì - ³
= í î
ì ³
= í î
38 Chap. 10
39 Chap. 10
Theorem 10.4’ (m-th arrival)
1 2
: the time at which the m-th event occurs
where ' are the i.i.d. exponential inter-arrival times. For the exponential r.v.
( )
( )
This implies that
m
m
m m
j
X
m S
m
S
S X X X
X s X
j
j S
w l
l w w l
l w
= + + +
F =
-
ì ü
F = í ý
î - þ
L
1
is an m-Erlang random variable whose pdf is given by ( )
( ) , 0.
( 1)!
m
m
y S
f y y e y
m l l
l
-
= - ³
-
1 2
: the time at which the m-th event occurs
where ' are the i.i.d. exponential inter-arrival times. For the exponential r.v.
( )
( )
This implies that
m
m
m m
j
X
m S
m
S
S X X X
X s X
j
j S
w l
l w w l
l w
= + + +
F =
-
ì ü
F = í ý
î - þ
L
1
is an m-Erlang random variable whose pdf is given by ( )
( ) , 0.
( 1)!
m
m
y S
f y y e y
m l l
l
-
= - ³
-
40 Chap. 10
Theorem 10.4’’ (arrival time of a single event)
( )
: arrival time of the single event that occurred in [0,t]
[ ( ) 1 and ( ) 1]
[ ] [ ( ) 1| ( ) 1]
[ ( ) 1]
[ ( ) 1] [ ( ) ( ) 0]
.
[ ( ) 1]
This imp
x t x
t
T
P N x N t
P X x P N x N t
P N t
P N x P N t N x xe e x
P N t te t
l l
l
l l
- - -
-
= =
£ = = = =
=
= - =
= = =
=
lies that the random occurrence (event) is uniformly distributed in (0,t).T
( )
: arrival time of the single event that occurred in [0,t]
[ ( ) 1 and ( ) 1]
[ ] [ ( ) 1| ( ) 1]
[ ( ) 1]
[ ( ) 1] [ ( ) ( ) 0]
.
[ ( ) 1]
This imp
x t x
t
T
P N x N t
P X x P N x N t
P N t
P N x P N t N x xe e x
P N t te t
l l
l
l l
- - -
-
= =
£ = = = =
=
= - =
= = =
=
lies that the random occurrence (event) is uniformly distributed in (0,t).T
41 Chap. 10
42 Chap. 10
43 Chap. 10
44 Chap. 10
45 Chap. 10
46 Chap. 10
{ }
( )
( )
1
2
, ( )
( )
i
i i
i i
S H T
N t S H
N t S T
=
= =
= =
å å
(
( (
47 Chap. 10
48 Chap. 10
49 Chap. 10
50 Chap. 10
51 Chap. 10
52 Chap. 10
53 Chap. 10
54 Chap. 10
55 Chap. 10
56 Chap. 10
57 Chap. 10
58 Chap. 10
Proof of Theorem 10.8
( ) ( )
( )
( )
2
1
1 1
1 1
2 1 2
1 2
/ 2 1
Let , 1, , .
That is,
.
( ) 1 .
2
1 1 1
;
1 1
n n n
n n n n n
k k
x t t
X
n n
X W t W t W W n k
W X
W X X
W X X X
f x e
t t
W AX A
a
pa
-
- -
é ù
- ë - û
-
= - º - =
=
= +
= + + +
=
-
é ù
ê ú
ê ú
= =
ê ú
ê ú
ë û
L
M
L
M O
L L
.
( ) ( )
( )
( )
2
1
1 1
1 1
2 1 2
1 2
/ 2 1
Let , 1, , .
That is,
.
( ) 1 .
2
1 1 1
;
1 1
n n n
n n n n n
k k
x t t
X
n n
X W t W t W W n k
W X
W X X
W X X X
f x e
t t
W AX A
a
pa
-
- -
é ù
- ë - û
-
= - º - =
=
= +
= + + +
=
-
é ù
ê ú
ê ú
= =
ê ú
ê ú
ë û
L
M
L
M O
L L
.
59 Chap. 10
[ ]
[ ] [ ]
[ ]
[ ]
[ ] [ ]
( ) ( )
1 1 0
1
1
1
2 1
1
1
1
1
0
0 0
( ) 1 5.11
| det |
det 1;
( ) ( )
( ) ( )
n
n
n n n
W X
k k
k
X X n
n
W X n n
E X E W W
E X
E X E W W
E W AE X
f w f A w refer to Theorem
A
w
w w
A A w
w w
f x f x
f w f w w
-
-
-
-
=
-
é ù
é ù - é ù
ê ú
ê ú ê ú
= ê ú = ê ú = ê ú
ê ú
ê ú - ê úë û
ë û ë û
= =
=
é ù
ê - ú
ê ú
= =
ê ú
ê ú
ë - û
=
= -
Õ
M M M
M
( )
( )
2
1 1
( ) / 2
1 1 1
1 .
2
n n n n
k k
w w t t
n n n n
e t t
a
pa
- é - ù
- - ë - û
= = -
=
Õ Õ
-[ ]
[ ] [ ]
[ ]
[ ]
[ ] [ ]
( ) ( )
1 1 0
1
1
1
2 1
1
1
1
1
0
0 0
( ) 1 5.11
| det |
det 1;
( ) ( )
( ) ( )
n
n
n n n
W X
k k
k
X X n
n
W X n n
E X E W W
E X
E X E W W
E W AE X
f w f A w refer to Theorem
A
w
w w
A A w
w w
f x f x
f w f w w
-
-
-
-
=
-
é ù
é ù - é ù
ê ú
ê ú ê ú
= ê ú = ê ú = ê ú
ê ú
ê ú - ê úë û
ë û ë û
= =
=
é ù
ê - ú
ê ú
= =
ê ú
ê ú
ë - û
=
= -
Õ
M M M
M
( )
( )
2
1 1
( ) / 2
1 1 1
1 .
2
n n n n
k k
w w t t
n n n n
e t t
a
pa
- é - ù
- - ë - û
= = -
=
Õ Õ
-60 Chap. 10
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