Biomedical Engineering Reference
In-Depth Information
since the sums of the
x
i
-
x
and
y
i
-
y
over all
i
in Eq. (E.36) are zero, by definition
of the mean values. Thus, the mean value of
Q
is the value of the function
Q
(
x
,
y
)
calculated at
x
=
x
and
y
=
y
.
The variance of the
Q
i
is given by
N
Q
i
-
Q
2
.
1
N
2
Q
σ
=
(E.38)
i
=
1
Applying Eq. (E.36) with
Q
=
Q
(
x
,
y
)
,wefindthat
∂
Q
∂
y
(
y
i
-
y
)
2
N
1
N
x
(
x
i
-
x
)+
∂
Q
2
Q
σ
=
(E.39)
∂
i
=
1
∂
2
1
N
x
)
2
+
∂
2
1
N
N
N
Q
∂x
Q
∂y
y
)
2
¯
=
(
x
i
-
¯
(
y
i
-
i
=
1
i
=
1
+2
∂
Q
∂
∂
Q
∂
1
N
N
(
x
i
-
x
)(
y
i
-
y
).
(E.40)
x
y
i
=
1
The last term, called the
covariance
of
x
and
y
, vanishes for large
N
if the values of
x
and
y
are uncorrelated. (The factors
y
i
-
x
are then just as likely to be
positive as negative, and the covariance also decreases as
1/
N
). We are left with the
first two terms, involving the variances of the
x
i
and
y
i
:
y
and
x
i
-
∂
2
x
+
∂
2
Q
∂
Q
∂
2
Q
2
2
y
.
σ
=
σ
σ
(E.41)
x
y
This is one form of the error propagation formula, which is easily generalized to a
function
Q
of any number of independent random variables.
