Information Technology Reference
In-Depth Information
define the mean value
E
(
g
◦
x
)
by the formula
(
◦
)=
E
g
x
gdF
R
if the integral exists.
Example 3.2.
Let
x
be discrete, i.e. there exist
x
i
∈
R
,
p
i
∈
(
0, 1
]
,
i
=
1, ...,
k
such that
)=
x
i
<
t
p
i
.
(
F
t
Then
k
i
=
1
x
i
p
i
.
The second classical case is the continuous distribution, where
E
(
x
)=
tdF
(
t
)=
R
t
−
∞
ϕ
(
(
)=
)
F
t
u
du
.
Then
∞
(
)=
(
)=
ϕ
(
)
E
x
tdF
t
t
t
dt
.
−
∞
R
Example 3.3.
Let us compute the dispersion
2
(
)=
(
◦
)
σ
x
E
g
x
,
where
2
,
a
(
)=(
−
)
=
(
)
g
u
u
a
E
x
.
Here we have two possibilities. The first
2
2
dF
σ
=
R
(
t
−
a
)
(
t
)
i.e.
k
i
=
1
(
x
i
−
a
)
2
2
p
i
σ
(
x
)=
in the discrete case, and
∞
−
∞
(
2
2
(
)=
−
)
ϕ
(
)
σ
x
t
a
t
dt
in the continuous case. The second possibility is the equality
2
2
x
2
a
2
(
)=
((
−
)
)=
(
)
−
(
)+
(
)=
σ
x
E
x
a
E
2
aE
x
E
x
2
a
2
,
a
=
(
)
−
=
(
)
E
E
x
.
x
2
t
2
, hence
=
(
)
(
)
(
)=
Since
a
E
x
is known, it is sufficient to compute
E
. In the case we have
g
t
x
2
t
2
dF
E
(
)=
g
(
t
)
dF
(
t
)=
(
t
)
.
R
R
