## Calculate covariance matrix of two given numbers of events

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.
1. The problem statement, all variables and given/known data
Variables $N_1[\itex] and [itex]N_2[\itex], distributed on [itex]y \in [0, 1][\itex] as follows: [itex]f_1(y<y_0) = 0; f_1(y>= y_0) = \frac{1}{1-y_0}$
[itex]f_2 = 1[\itex]
Define: [itex]N_{<} = [\itex]amount of events falling below [itex]y_0[\itex], [itex]N_{>}[\itex] analogously...
2. Relevant equations
Then, we can calculate:
[itex]N_2 = \frac{1}{y_0}N_{<}[\itex]
[itex]N_1 = -\frac{1-y_0}{y_0}N_{<} + N_{>}[\itex]
Covariance of two variables: [itex]Cov(a,b) = <(a·b)^2> - <a·b>^2[\itex]

3. The attempt at a solution
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].
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).
However, I cannot extend this to the off-diagonal terms.