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Hi, I am trying to follow this paper: (arXiv link).
On page 18, Appendix A.1, the authors calculate a covariance matrix for two variables in a way I cannot understand.
Variables [itex]N_1[\itex] <br /> and [itex]N_2[\itex], distributed on <br /> [itex]y \in [0, 1][\itex] as follows:<br /> [itex]f_1(y<y_0) = 0; f_1(y>= y_0) = \frac{1}{1-y_0}[/itex]<br /> [itex]f_2 = 1[\itex]<br /> Define: [itex]N_{<} = [\itex]amount of events falling below [itex]y_0[\itex], [itex]N_{>}[\itex] analogously...<br /> <h2>Homework Equations</h2><br /> Then, we can calculate:<br /> [itex]N_2 = \frac{1}{y_0}N_{<}[\itex]<br /> [itex]N_1 = -\frac{1-y_0}{y_0}N_{<} + N_{>}[\itex]<br /> Covariance of two variables: [itex]Cov(a,b) = <(a·b)^2> - <a·b>^2[\itex]<br /> <br /> <h2>The Attempt at a Solution</h2><br /> After calculating the two previous results myself (N1 and N2), I tried to calculate the covariance matrix V (please see pdf). I tried to do it from the definition of covariance, but I didn't get anywhere (or to different results). So I decided to guess if the authors were doing the usual error propagation from [itex]N_1, N_2[\itex] as a function of [itex]N_{<}, N_{>}[\itex].<br /> Considering the usual in physics for N large, [itex]\sigma(N_k) = \sqrt(N_k), k \in {<, >}[\itex], and then, for instance [itex]\sigma^2(N_2) =\sum_k \frac{\partial N_2}{\partial N_k} \sigma(N_k) = \frac{1}{y_0 ^2} N_{>}[\itex] (the right result).<br /> However, I cannot extend this to the off-diagonal terms.<br /> <br /> Could somebody please help me? Thanks!<br /> <br /> EDIT: I cannot see the latex expressions, and instead I see all my itex \itex ! How can I solve this? Thanks![/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex]
On page 18, Appendix A.1, the authors calculate a covariance matrix for two variables in a way I cannot understand.
Homework Statement
Variables [itex]N_1[\itex] <br /> and [itex]N_2[\itex], distributed on <br /> [itex]y \in [0, 1][\itex] as follows:<br /> [itex]f_1(y<y_0) = 0; f_1(y>= y_0) = \frac{1}{1-y_0}[/itex]<br /> [itex]f_2 = 1[\itex]<br /> Define: [itex]N_{<} = [\itex]amount of events falling below [itex]y_0[\itex], [itex]N_{>}[\itex] analogously...<br /> <h2>Homework Equations</h2><br /> Then, we can calculate:<br /> [itex]N_2 = \frac{1}{y_0}N_{<}[\itex]<br /> [itex]N_1 = -\frac{1-y_0}{y_0}N_{<} + N_{>}[\itex]<br /> Covariance of two variables: [itex]Cov(a,b) = <(a·b)^2> - <a·b>^2[\itex]<br /> <br /> <h2>The Attempt at a Solution</h2><br /> After calculating the two previous results myself (N1 and N2), I tried to calculate the covariance matrix V (please see pdf). I tried to do it from the definition of covariance, but I didn't get anywhere (or to different results). So I decided to guess if the authors were doing the usual error propagation from [itex]N_1, N_2[\itex] as a function of [itex]N_{<}, N_{>}[\itex].<br /> Considering the usual in physics for N large, [itex]\sigma(N_k) = \sqrt(N_k), k \in {<, >}[\itex], and then, for instance [itex]\sigma^2(N_2) =\sum_k \frac{\partial N_2}{\partial N_k} \sigma(N_k) = \frac{1}{y_0 ^2} N_{>}[\itex] (the right result).<br /> However, I cannot extend this to the off-diagonal terms.<br /> <br /> Could somebody please help me? Thanks!<br /> <br /> EDIT: I cannot see the latex expressions, and instead I see all my itex \itex ! How can I solve this? Thanks![/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex][/itex]
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