Digital Signal Processing Reference
In-Depth Information
This indicates that we can use a Poisson approximation (
5.143
):
000
g¼
1
X
3
3
k
k
!
e
3
Pfk>
3
3
;
000
g¼
1
Pfk
3
3
;
¼
1
0
:
6472
;
;
k¼
0
¼
0
:
3528
:
(5.269)
Exercise 5.17
An electronic system has 2,000 components. In a given period of
time, the probability of failure of any one of its components is equal to 0.001 and is
not dependent on the failure of any other components. Find the probability that
exactly three components will fail.
Answer
10
9
2
;
000
3
ð
1
10
3
2
;
000
3
Pf
3
;
2
;
000
g¼
Þ
2
;
000
!
10
9
ð
1
10
3
1
;
997
¼
Þ
¼
0
:
1805
:
(5.270)
1
;
997
3
!
!
Let us now find the approximated probability (
5.270
) using a Poisson approxi-
mation (
5.143
), finding that the following condition it is satisfied:
000
Þð
10
Þ
3
np ¼ð
2
;
¼
2
;
(5.271)
2
3
3
e
2
Pfk ¼
3
g¼
¼
0
:
1804
:
(5.272)
!
Note that (
5.272
) is a good approximation of (
5.270
).
Exercise 5.18
In a random sequence of binary symbols “0” and “1,” see, for
example,
...
00000111010001111100
...
(5.273)
the probabilities of “0” and “1” are equal and independent of one another.
A successive sequence of identical symbols forms groups, as shown below:
00000
111
0
1
000
11111
00
...
:
(5.274)
...
There are a total of
n
groups in a sequence.
Define a random variable
X
as a number of equal symbols in
i
th group. Then
define a random variable
Y
as a ratio of the number of symbols in all groups divided
by the number of groups
n
. Find the mean value of
Y
.
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