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Homework Help: Calculate covariance matrix of two given numbers of events

  1. Apr 9, 2012 #1
    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 [itex]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]
    [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.

    Could somebody please help me? Thanks!

    EDIT: I cannot see the latex expressions, and instead I see all my itex \itex ! How can I solve this? Thanks!
    Last edited: Apr 9, 2012
  2. jcsd
  3. Apr 9, 2012 #2
    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
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