Biomedical Engineering Reference
In-Depth Information
or for discrete distributions as given by (7.19).
e
tx
P
(
x
)
≡
e
tx
=
M
(
t
)
(7.19)
≡
e
tx
is the moment-generating function. Expansion of (7.18) yields
(7.20), the general moment-generating function.
where
M
(
t
)
∞
1
1
2!
t
2
x
2
3!
t
3
x
3
M
(
t
)
=
(1
+
tx
+
+
+···
)
P
(
x
)
dx
−∞
(7.20)
1
1
1
2!
t
2
m
2
+
3!
t
3
m
3
+···+
r
!
t
r
m
r
=
1
+
tm
1
+
where
m
r
is the
r
th moment about zero.
For independent variables
x
and
y
, the moment-generating function may be used
to generate joint moments, as shown in (7.21).
e
t
(
x
+
y
)
=
e
tx
e
ty
=
M
x
+
y
(
t
)
=
M
x
(
t
)
M
y
(
t
)
(7.21)
If
M
(
t
) is differentiable at zero, the
r
th moments about the origin are given by the
following equations (7.22):
=
e
tx
,
M
(
t
)
which is
M
(0)
=
1
=
xe
tx
,
M
(
t
)
which is
M
(0)
=
x
=
x
2
e
tx
,
=
x
2
M
(
t
)
which is
M
(0)
(7.22)
=
x
3
e
tx
,
=
x
3
M
(
t
)
which is
M
(0)
=
x
r
e
tx
,
=
x
r
r
(
t
)
r
(0)
M
which is
M
Therefore, the mean and variance of a distribution are given by equations in (7.23).
M
(0)
μ
≡
x
=
(7.23)
≡
x
2
−
−
M
(0)
2
2
2
M
(0)
σ
x
=
.
7.5 SUMMARY
The moments may be simply computed using the moment-generating function. The
n
th
raw or first moment, that is, moment about zero; the origin (
μ
n
) of a distribution
P
(
x
)
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