1.2.
PROBABILITY MEASURES
15
(
b
) If we assume to have two coins, i.e.
Ω =
{
(
T, T
)
,
(
H, T
)
,
(
T, H
)
,
(
H, H
)
}
and
F
= 2
Ω
, then the intuitive assumption ’fair’ leads to
P
(
{
ω
}
) :=
1
4
.
That means, for example, the probability that exactly one of two coins
shows heads is
P
(
{
(
H, T
)
,
(
T, H
)
}
) =
1
2
.
Example 1.2.4
[Dirac and counting measure]
2
(
a
)
Dirac measure
: For
F
= 2
Ω
and a fixed
x
0
∈
Ω we let
δ
x
0
(
A
) :=
1
:
x
0
∈
A
0
:
x
0
∈
A
.
(
b
)
Counting measure
: Let Ω :=
{
ω
1
, ..., ω
N
}
and
F
= 2
Ω
. Then
μ
(
A
) := cardinality of
A.
Let us now discuss a typical example in which the
σ
–algebra
F
is not the set
of
all
subsets of Ω.
Example 1.2.5
Assume that there are
n
communication channels between
the points
A
and
B
. Each of the channels has a communication rate of
ρ >
0
(say
ρ
bits per second), which yields to the communication rate
ρk
, in case
k
channels are used. Each of the channels fails with probability
p
, so that
we have a random communication rate
R
∈ {
0
, ρ, ..., nρ
}
. What is the
right
model for this? We use
Ω :=
{
ω
= (
ε
1
, ..., ε
n
) :
ε
i
∈ {
0
,
1
}
)
with the interpretation:
ε
i
= 0 if channel
i
is failing,
ε
i
= 1 if channel
i
is
working.
F
consists of all unions of
A
k
:=
{
ω
∈
Ω :
ε
1
+
· · ·
+
ε
n
=
k
}
.
Hence
A
k
consists of all
ω
such that the communication rate is
ρk
.
The
system
F
is the system of observable sets of events since one can only observe
how many channels are failing, but not which channel fails. The measure
P
is given by
P
(
A
k
) :=
n
k
p
n

k
(1

p
)
k
,
0
< p <
1
.
Note that
P
describes the
binomial distribution
with parameter
p
on
{
0
, ..., n
}
if we identify
A
k
with the natural number
k
.
2
Paul Adrien Maurice Dirac, 08/08/1902 (Bristol, England)  20/10/1984 (Tallahassee,
Florida, USA), Nobelprice in Physics 1933.