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