add explanation what the effective livetime is

docs/livetime.rst

@@ -12,6 +12,20 @@ the sum of the weights.
 
     L_{eff} = \frac{\sum{w_i}}{\sum{w_i^2}}
 
+The effective livetime is defined as the time the detector would need to take data to have a sample
+with the same average, relative uncertainty as the MC set [#]_.
+The uncertainty of an unweighted sample with :math:`N=RL` events is :math:`\sqrt{N}`,
+which assumes a Poisson distribution
+of the event number. :math:`R` is the event rate and :math:`L` the livetime.
+The uncertainty of the event rate :math:`R=\sum_i w_i` from a weighted MC set is
+:math:`\Delta R = \sqrt{\sum_i w_i^2}` [#]_.
+Thus we can easily derive the above definition from
+
+.. math::
+    \frac{\sqrt{N}}{N} = \frac{\sqrt{RL}}{RL} \overset{!}{=} \frac{\Delta R}{R}
+
+Rearranging the last equality results in the above expression for the effective livetime.
+
 
 As shown in the example below it can be calculated for any sample or for any subsample by using histograms.
 
@@ -20,3 +34,7 @@ As shown in the example below it can be calculated for any sample or for any sub
 .. figure:: livetime_rate.svg
 
 .. figure:: livetime_livetime.svg
+
+
+.. [#] See `this <https://internal-apps.icecube.wisc.edu/reports/data/icecube/2009/02/001/icecube_200902001_v2.pdf>`_ report by Michelangelo D’Agostino (requires password access).
+.. [#] This can be derived from the fundamental properties of the variance: :math:`\sigma^2(cx)=c^2\sigma^2(x)` and :math:`\sigma^2(x_1+x_2)=\sigma^2(x_1)+\sigma^2(x_2)`, see e.g. `here <https://www.boost.org/doc/libs/1_83_0/libs/histogram/doc/html/histogram/rationale.html#histogram.rationale.weights>`_.