error propagation for Endcap background subtraction
Lets define all:
A is # of eve accepted w/o Endcap ~~~(1)
B is # of eve rejected if Endcap is used
C is # of eve accepted if Endcap is used
X is estimator of # of eve accepted if East \& West Endcap were used.
Obviously B+C=A ; B \& C are statistically independent,
so
~~$\sigma(A)=\sqrt(B+C)$; ~~$\sigma(B)=\sqrt(B)$; ~~$\sigma(C)=\sqrt(C)$ ~~~(2)
The estimator X is approximated as
$X=C-B$ ~~~(3)
Lets define the signal/background ratio $R_
$, where as background we count all QCD events not discarded if neither Endcap is not used
$R_
=\frac
$ ~~~(4)
Since numerator \& denominator are correlated (via X -variable) lets do some substitutions:
$R_
=\frac
- \frac
$ ~~~(5)
Now we have ratio of statistically independent variables and it is easy to compute:
~~ $\sigma(R_
)=\frac
\sqrt{\frac
+ \frac
{B}}$ ~~~(6)