Appendix A Algorithms
272
Equation 12 Two-Tailed Probability at x
−===
∫∫
−
−
−
−
σππ
2
x
erfc1dte
2
dte
1
)x(P
x
0
t
x
x
t
tailed2
22
(12)
… where erfc is the complementary error function defined in Equation 13:
Equation 13 Complimentary Error Function
(13)
The error and complementary error functions are important special functions that are
widely used in the probability theory; there is a direct connection between the error
and complimentary error functions and normalized Gaussian function.
For a “two-tailed” distribution, the count rate value is expected to fall within the area
defined between
to
. As for the one tailed distribution, only positive data
values are possible, although at low count rates (i.e., those resulting from ambient
background conditions) the calculated range can overlap zero.
The one-tailed probability is a special case of two-tailed probability when
Equation 14 One-Tailed Probability
( )
∫
+
∞−
−
−
−
=+<<−∞
σµ
σ
µ
σπ
σµ
dxe
2
1
)x(P
2
2
2
x
tailed1
(14)
The one-tailed Gaussian integral gives the probability that a point lies within the
given number of σ to one side of the mean.
Figure 129: One-tailed distribution
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