Appendix
========

The derivation of the segment-specific readnoise variance (:math:`{ var^R_{s}  }`)
is shown here. This pertains to both the 'conventional' and 'weighted' readnoise
variances - the only difference being the number of groups in the segment.
This derivation follows the standard procedure for fitting data to a straight line,
such as in chapter 15 of Numerical Recipes.  The segment-specific variance from
read noise corresponds to :math:`{\sigma_b^2}` in section 15.2. 

For read noise R, weight w = :math:`{1 / R^2}`, which is a constant. 
  
n = number of groups (`ngroups` in the text)

t = group time (`tgroup` in the text)

x = starting time for each group, = :math:`{(1,2,3, ... n+1) \cdot t}`


:math:`{S_1 = \sum_{k=1}^n w}`

:math:`{S_x = \sum_{k=1}^n (w  \cdot  x_k) t}`

S\ :sub:`xx`\  = :math:`{\sum_{k=1}^n (w \cdot x_k)^2 t^2}`

D = :math:`{S_1 \cdot S}`\ :sub:`xx`\ - :math:`{S_x^2}`


Summations needed:

:math:`{\sum_{k=1}^n k = n \cdot (n+1) / 2 = n^2 /2 + n/2 }`

:math:`{\sum_{k=1}^n k^2= n \cdot (n+1) \cdot (2 \cdot n+1) / 6 = n^3/3 + n^2/2 +n/6 }`

      
The variance due to read noise 
= :math:`{var^R_{s} = S_1 / D = S_1 / (S_1 \cdot S_{xx} - S_x^2)}` 


= :math:`{ \dfrac {w \cdot n} { [w \cdot n \cdot \sum_{k=1}^n (w \cdot x_k^2 \cdot t^2)] - [\sum_{k=1}^n (w \cdot x_k \cdot t)] ^2}}`     


= :math:`{ \dfrac {n} { w \cdot t^2 \cdot [ n \cdot ( n^3/3 + n^2/2 +n/6 ) - (n^2/2 + n/2 )^2 ] }}`
    

= :math:`{ \dfrac {1} { ( n^3/12 - n/12 ) \cdot w \cdot t^2 }}`


= :math:`{ \dfrac{12 \cdot R^2}  {(n^3 - n) \cdot t^2}}` 

This is the equation used in the step code and listed in the segment-specific computations
section of the step Description.