Mirion Technologies Lynx II DSA - Area Uncertainty Calculation

Appendix I Non-parameterized ROI
196 Lynx II DSA User's Manual - 7096089
where
(7)
∑
=
=
i
1j
ji
YP
Area Uncertainty
The standard deviation (at one sigma) of the net peak area is calculated from the equation
(8)
2
B
2
Gs
σ+σ=σ
To establish σB, let us consider a function, F, which is some combination of counts in
several channels.
(9)
( )
n21
y,,y,yfF =
where y
1
, ≤, y
n
are counts in n channels.
Assuming that the y
i
’s are uncorrelated, which is the case in gamma spectroscopy, the
estimate of the variance of F is given by
1
:
(10)
2
y
2
n
2
y
2
2
2
y
2
1
2
F
n21
y
f
y
f
y
f
σ








∂
∂
+σ








∂
∂
+σ








∂
∂
=σ 
Using this relationship, the variance of a linear continuum becomes
(11)
2
B
2
2
B
2
2
B
21
n2
N
n2
N
σ






+σ






=σ
where
2
B
1
σ
is the variance of B1, and
2
B
2
σ
is the variance of B2.
Making use of the properties of Poisson distributed quantities, and combining the terms,
equation (36) can be written as:
(12)
( )
21
2
2
B
BB
n2
N
+






=σ
1
A poof of this may be found in a variety of textbooks that discuss error propagation. In the exact case, the covariances must
be included. However, the covariance terms vanish, if the N’s are uncorrelated, as is the case in gamma spectroscopy.