Biomedical Engineering Reference
In-Depth Information
Fig. 11.5
Probability density
P
n
(
r
n
) for net count rate
r
n
.When
activity
A
=
0,
r
1
is fixed by choice of probability
α
for type-I
errors. Use of
r
1
and choice of probability
β
for type-II errors
fixes
r
2
, corresponding to the minimum detectable true activity
A
II
.(If
α
=
β
, then the intersection of the two curves occurs at
the value
r
n
=
r
1
.) (Courtesy James S. Bogard.)
Repeated measurements of the gross and background count rates with the ac-
tivity
A
II
in a sample would give the net count rate,
r
g
-
r
b
, distributed about
r
2
,
as shown in Fig. 11.5. Since the quantity
r
g
-
r
b
-
r
2
is distributed normally about
the mean value of zero, we can describe it with the help of the standard normal
distribution. Letting
k
β
represent the number of standard deviations that leave an
area
β
to its left, we write [analogous to Eq. (11.67)]
r
g
t
g
+
r
b
r
g
-
r
b
-
r
2
=
-
k
β
(11.75)
t
b
r
g
-
r
b
t
g
+
r
b
t
g
+
t
b
t
g
t
b
,
= -
k
β
(11.76)
where
r
b
/
t
g
has been subtracted and added under the radical. On both sides of this
equation we set the net count rate
r
g
-
r
b
=
r
1
, the decision-level rate established for
type-I errors. We can now write
r
1
t
g
+
r
b
t
g
+
t
b
.
r
2
=
r
1
+
k
β
(11.77)
t
g
t
b
Substituting for
r
1
from Eq. (11.68), we obtain
k
α
k
2
α
t
g
+4
r
b
t
g
+
t
b
2
t
g
+
1
r
2
=
k
α
2
t
g
t
b
