It's not a variable, just that sensor size affects the density of light given that the same total amount of light falls on the sensor. On the larger sensor, the light is refracted to fill a larger frame, thus the light is less dense.
So ... Are you saying that f/8, 1/125 will give a different exposure on full frame than it will on crop, with both cameras side-by-side, in the same light and the same lens?
edit
And same ISO, obviously...
(If that is what you are saying (which is what it sounds like to me) - you are wrong.)
No, that's not what I mean. A f/8, 1/125 and at the same ISO will give the same exposure for FX and DX, because their pixel pitch is the same. If both FX and DX have the same amount of pixels, DX would have about 1 time smaller pixels. So, at base ISO for both sensors with the same design, without amplification, DX's base ISO would be 1 stop less, the exposure for both would be different. Even at the same amount of amplification, DX's ISO would always be about 1 stop less.
This makes FX has higher low light capabilities - at the same f/stop. But that is not true, cause' FX's lenses would always be larger than DX's lenses at the same f/stop, which means FX basically gets more light because their lens. If both DX and FX's lens were at the same size, both DX and FX would get the same amount of light and their low light capabilities would be the same. A DX's effective f/stop is one stop less compared to FX. It's the lens that controls the low light ability, not the sensor. Do you realise why compact cameras are such bad low light performers? Cause' their lenses are small.