The noise is supposedly Poissonian (in electrons, not in counts).
Hence it is easier to transform counts to electrons to compute the sigma of the noise.

Naming convention: 
  I<sub>p</sub> means image in pixel space, i.e. where the values are in counts;
  I<sub>e</sub> instead is the corresponding image where values are in electrons.
Remember simple formula:

   if     y = k * x     then       var(y) = k<sup>2</sup> * var(x)


Single image I:

     Poisson =&gt;   var(I<sub>e</sub>)   =   I<sub>e</sub>   =   g * I<sub>p</sub> 

     Hence:

        I<sub>p</sub> = I<sub>e</sub> / g
     and
        var(I<sub>p</sub>) = var(I<sub>e</sub>) / g<sup>2</sup>   =   I<sub>p</sub> / g

     Eff. Gain := g
</PRE>



<HR width="80">


<PRE>
Coadded images T = sum( I<sub>i</sub> ):

     Poisson =&gt;   var(T<sub>e</sub>)   =   T<sub>e</sub>   =   sum( I<sub>e</sub><sub>i</sub> )   =    g * sum( I<sub>p</sub><sub>i</sub> )   =   g * T<sub>p</sub>

     hence: => 

        T<sub>p</sub> = T<sub>e</sub> / g   
        var(T<sub>p</sub>)   =   var(T<sub>e</sub>) / g<sup>2</sup>   =   T<sub>e</sub> / g<sup>2</sup>   =   T<sub>p</sub> / g

     Eff. Gain := g
</PRE>



<HR width="80">



<PRE>
Averaged image M = sum( I<sub>i</sub> ) / N :

Here though the poissonian statistics <B>cannot</B> be applied to sum( I<sub>e</sub><sub>i</sub> ) / N !

In fact, the poissonian statistics applies to the total number of electrons.

     Poisson =&gt;   var(T<sub>e</sub>)   =   T<sub>e</sub>   =   sum( I<sub>e</sub><sub>i</sub> )    =    g * sum( I<sub>p</sub><sub>i</sub> )    =   g * N * M<sub>p</sub>

                     T = N * M
     Poisson =&gt;   var(T<sub>e</sub>)   =   T<sub>e</sub>
      Hence
                     var(N * M<sub>e</sub>)   =   N * M<sub>e</sub>

                     N<sup>2</sup> * var(M<sub>e</sub>)   =   N * M<sub>e</sub>

             (1)     var(M<sub>e</sub>)   = M<sub>e</sub> / N

       and since
             (2)     M<sub>e</sub> = M<sub>p</sub> * g

             (3)     var(M<sub>e</sub>) = var(M<sub>p</sub>) * g<sup>2</sup>

       given 1,2,3:

                     M<sub>e</sub> / N = M<sub>p</sub> * g / N = var(M<sub>p</sub>) * g<sup>2</sup>

                     var(M<sub>p</sub>) = M<sub>p</sub> / ( N * g )


     Eff. Gain := N * g
</PRE>