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Interesting discussions...
My "Bible" "Handbook of CCD Astronomy" by Steve Howell, covers the whole gambut of noise and noise generation (p 50 - 82)
on the effects of On Chip binning he says-
"Binning of CCD pixels decreases the image resolution, usually increases the final SNR....and reduces the total readout time"
"we would get a final signal level equal to ~4 times each single pixel value, but only one times the read noise"
He explains (p75-p82) in painful detail the "CCD Equation" for SNR calculation used by the professional astronomers around the world.
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