PCM-2 Manual.pdf - Voss Associates

In Figure 6 the shaded area corresponds to the Gaussian Cumulative Distribution operating on SF -­
mathematically equal to equation 31. The distribution represents the probability that the background
count rate will be less than RA(MIN)' so the false alarm rate is determined by Eq. 36 where F(SF) is
short-hand notation for the Gaussian Cumulative Distribution Function operating on SF.
p = 1 - F(SF)
Eq. 36
EXAMPLE: In light of earlier comments, the false alarm rate is controlled entirely by SF, and is
independent of R B • (If Table 2 is invoked, the independent variable ... "x" is substituted for "SF."
It is noted that if SF is equal to 3.09. the table indicates F(SF) is equal to 0.9989991 and p = 1 ­
F(SF) = 0.0010009.
Consistent with Eq. 33, in the case of N channels or N measurements, the overall false alarm
probability is indicated by Eq. 37.
P = 1 - F(SF)N
Eq. 37
Because F(SF) is always less than 1, raising it to a power greater than 1 will reduce the value and
increase the overall false alarm probability. This suggests that with an increasing number of
channels, larger Sigma Factors are required to maintain an acceptable false alarm rate. Larger Sigma
Factors decrease sensitivity as will be discussed later in this writing, so some compromise is always
in order.
Table 2 can be used in inverse fashion by entering its field with p and extracting SF. This is the
more logical approach to using the table. Eq. 38 expresses this technique mathematically where pI is
notation for "the inverse function of F."
Eq. 38
The process of invoking Eq. 38 and entering Table 2 with the results is greatly simplified by using
the Sigma Factor Nomograph illustrated by Figure 7. To use the nomograph, extend a straight line
from the tick mark on the left-hand column that corresponds to the number of channels in use through
the desired false alann probability in the middle column until it intersects the right-hand column. The
required Sigma Factor will be indicated. In analogous fashion. any of the three variables can become
the dependent variable.
PCM2.MAN