Expression of a variance with shot noise

In an Astrophysics context, basically, I have an expression of the standard deviation of a quantity (spherical harmonics C_l) which has a noise. I am interested in the variance expression of an estimator which includes the integrated (actually summed) quantity and the integrated shot noise (also actually integrated). I want to be sure that part of integrated shot noise has the same pre-factor than for the variance part of integrated quantity.