Biomedical Engineering Reference
In-Depth Information
=
Since, for small
y
,
e
y
1+
y
, the series of factors in the denominator can be rewrit-
ten (
µ
is large) to give
1
1
√
2
e
-(1+2+···+
x
)/
µ
.
P
x
=
√
2
e
x
/
µ
=
(E.33)
e
1/
µ
e
2/
µ
···
πµ
πµ
Thesumofthefirst
x
positive integers, as they appear in the exponent, is
x
(1 +
x
)/2 = (
x
2
+
x
)/2
=
x
2
/2
, where
x
has been neglected compared with
x
2
.Thus,we
find that
1
√
2
e
-
x
2
/2
µ
.
P
x
=
(E.34)
πµ
This function, which is symmetric in
x
, represents an approximation to the Pois-
son distribution. The
n
ormal distribution is obtained when we replace the Poisson
standard deviation
√
µ
by an independent parameter
σ
and let
x
be a continuous
random variable with mean value
µ
(not necessarily zero). We then write for the
probability density in
x
(-
∞
<
x
<
∞
)
the normal distribution
1
√
2
πσ
e
-(
x
-
µ
)
2
/2
σ
2
,
(E.35)
f
(
x
)
=
with
σ
2
>0. It can be shown that this density function is normalized (i.e., its inte-
gral over all
x
is unity) and that its mean and standard deviation are, respectively,
µ
and
σ
. The probability that the value of
x
lies between
x
and
x
+d
x
is
f
(
x
)d
x
.
Whereas the Poisson distribution has the single parameter
µ
, the normal distribu-
tion is characterized by the two independent parameters,
µ
and
σ
.
Error Propagation
We determine the standard deviation of a quantity
Q
(
x
,
y
)
that depends on two inde-
pendent, random variables
x
and
y
. A sample of
N
measurements of the variables
yields pairs of values,
x
i
and
y
i
,with
i
,
N
. For the sample one can compute
=
1, 2,
...
the means, ¯
Q
(
x
i
,
y
i
).
We assume that the scatter of the
x
i
and
y
i
about their means is small. We can then
write a power-series expansion for the
Q
i
about the point (
x
and
¯
y
; the standard deviations,
σ
x
and
σ
y
; and the values
Q
i
=
¯
y
), keeping only the
¯
x
,
first powers. Thus,
Q
(
x
,
y
)+
∂
Q
∂
x
(
x
i
-
x
)+
∂
Q
Q
i
=
Q
(
x
i
,
y
i
)
=
y
(
y
i
-
y
),
(E.36)
∂
where the partial derivatives are evaluated at
x
=¯
x
and
y
=
y
. The mean value of
Q
i
is simply
N
N
1
N
1
N
1
N
NQ
(
x
,
y
) =
Q
(
x
,
y
),
Q
≡
Q
i
=
Q
(
x
,
y
) =
(E.37)
i
=
1
i
=
1
