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