Digital Signal Processing Reference
In-Depth Information
Consequently, error results if we decide either that “1” was sent when “0” was
sent (
P
e
), or we decide that “0” was sent when “1” was sent (
P
e
). Due to the
symmetry of the PDFs both are equally probable:
P
e
0
¼ P
e
1
;
(5.250)
1
P
e
¼
5e
v
d
v ¼
0
5e
V=
2
0
:
:
:
(5.251)
V=
2
The total error is:
P
e
¼ Pð
0
ÞP
e
þ Pð
1
ÞP
e
¼
0
5e
V=
2
5e
V=
2
:
5
0
:
þ
0
:
5
0
:
5e
V=
2
¼
0
:
:
(5.252)
Exercise 5.12
The
n
random calls are made in a time interval [0,
T
]. What is the
probability that
k
of
n
calls occur within the time interval
t
(which is inside the
interval
T
).
Answer
Denote as the event as
A
if a call is received in
t
.
The probability of event
A
can be expressed as a ratio of the time
t
to the time
T
:
PfAg¼t=T:
(5.253)
The probability that
k
of
n
calls received in the time interval
t
is described using
the Bernoulli formula:
t
T
k
nk
n
k
t
T
Pfk
;
ng¼
1
:
(5.254)
Exercise 5.13
The binomial random variable
X
has the parameters
n ¼
5 and a
probability
p ¼
1/3. Find the expected value of the random variable
Y
,
Y ¼
e
2
X
:
(5.255)
Answer
The probability mass function of the variable
X
is given with the Bernoulli
formula:
1
3
k
5
k
5
k
2
3
PfX ¼ k
;
5
g¼
:
(5.256)
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