Calculate covariance matrix of two given numbers of events

[Brais]

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.

Homework Statement

Variables N_1[\itex] <br /> and N_2[\itex], distributed on &lt;br /&gt; y \in [0, 1][\itex] as follows:&amp;lt;br /&amp;gt; f_1(y&amp;amp;amp;lt;y_0) = 0; f_1(y&amp;amp;amp;gt;= y_0) = \frac{1}{1-y_0}&amp;lt;br /&amp;gt; f_2 = 1[\itex]&amp;amp;lt;br /&amp;amp;gt; Define: N_{&amp;amp;amp;amp;lt;} = [\itex]amount of events falling below y_0[\itex], N_{&amp;amp;amp;amp;amp;amp;gt;}[\itex] analogously...&amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;lt;h2&amp;amp;amp;amp;amp;gt;Homework Equations&amp;amp;amp;amp;amp;lt;/h2&amp;amp;amp;amp;amp;gt;&amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; Then, we can calculate:&amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;gt; N_2 = \frac{1}{y_0}N_{&amp;amp;amp;amp;amp;amp;amp;lt;}[\itex]&amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;gt; N_1 = -\frac{1-y_0}{y_0}N_{&amp;amp;amp;amp;amp;amp;amp;amp;lt;} + N_{&amp;amp;amp;amp;amp;amp;amp;amp;gt;}[\itex]&amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;gt; Covariance of two variables: Cov(a,b) = &amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;(a·b)^2&amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; - &amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;a·b&amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;^2[\itex]&amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;lt;h2&amp;amp;amp;amp;amp;amp;amp;amp;gt;The Attempt at a Solution&amp;amp;amp;amp;amp;amp;amp;amp;lt;/h2&amp;amp;amp;amp;amp;amp;amp;amp;gt;&amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;gt; 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&amp;amp;amp;amp;amp;amp;amp;amp;amp;#039;t get anywhere (or to different results). So I decided to guess if the authors were doing the usual error propagation from N_1, N_2[\itex] as a function of N_{&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;}, N_{&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;}[\itex].&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; Considering the usual in physics for N large, \sigma(N_k) = \sqrt(N_k), k \in {&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;, &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;}[\itex], and then, for instance \sigma^2(N_2) =\sum_k \frac{\partial N_2}{\partial N_k} \sigma(N_k) = \frac{1}{y_0 ^2} N_{&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt;}[\itex] (the right result).&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; However, I cannot extend this to the off-diagonal terms.&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; Could somebody please help me? Thanks!&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;lt;br /&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;gt; EDIT: I cannot see the latex expressions, and instead I see all my itex \itex ! How can I solve this? Thanks!

 

[Brais]

Right...
after some trial and error, I got to an expression for the right solution of the covariance matrix:

Given the (2x2) covariance matrix V, and the variables:

N1 = a1/c
N2 = -(1-c)a1/c + a2

I calculated V(m,n) = sum(i = 1 to 2) (dNm/da_i)(dNn/da_i) Error^2(Ni),
where Error^2(Ni) = Ni, and m, n are either 1 or 2.

This works and I get the right solution. Also, the expression makes sense... However, I couldn't find this in a book on statistics. Could somebody point me in a good direction? Do you know of any book/page were I can see the proof of this?

Thank you! And sorry for the wrong latex formulae in the previous post, but I don't know how to fix it. I counted the number of itex \itex and it is right :S