# Appendix¶

The derivation of the segment-specific readnoise variance ($${ var^R_{s} }$$) 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 to fitting data to a straight line, such as in chapter 15 of Numerical Recipes. The segment-specific variance from read noise corresponds to $${\sigma_b^2}$$ in section 15.2.

For read noise R, weight w = $${1 / R^2}$$, 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, = $${(1,2,3, ... n+1) \cdot t}$$

$${S_1 = \sum_{k=1}^n w}$$

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

Sxx = $${\sum_{k=1}^n (w \cdot x_k)^2 t^2}$$

D = $${S_1 \cdot S}$$xx- $${S_x^2}$$

Summations needed:

$${\sum_{k=1}^n k = n \cdot (n+1) / 2 = n^2 /2 + n/2 }$$

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

The variance from read noise = $${var^R_{s} = S_1 / D = S_1 / (S_1 \cdot S_{xx} - S_x^2)}$$

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

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

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

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

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