The reason why using an f1.8 lens wide open on crop gives a brighter image than f2.8 on FF (when both are at the same ISO and shutter speed) is the amplification of the crop cameras sensor is 2.56x greater, at the expense of noise at any given ISO rating.
What? That's wrong.
A crop f/1.8 lens and a full frame f/1.8 lens will provide exactly the same exposure when used at the same shutter speed and ISO. The full frame exposure WILL NOT be brighter.
You're right about an FF f/2.8 lens having more light gathering ability than a crop f/1.8 lens, but all that light it gathers is spread over a larger sensor, which makes the exposure more than a full stop darker than if you had used an f/1.8 lens.
So in terms of exposure, a f/1.8 lens is brighter than an f/2.8 lens, regardless of sensor size. Sensor size does affect depth of field, but that's a different story.
Exactly. Try it for yourself on a crop body and FF. f1.8 at 1/30 sec at 100 ISO will give you the same exposure on both cameras. FF will not be brighter.
Hey guys, read what I wrote. I said f1.8 on crop
is brighter than f2.8 on full frame when both have the same shutter speed and ISO, then you start telling me that I'm wrong to say FF is brighter when they're both at f1.8 and the same shutter speed and ISO. I didn't say that, so what gives? I went to great lengths to explain that at the same aperture, shutter speed and ISO,
they both expose the same due to the different light gathering of the format being compensated for by the amplifiers being set different. If you can look past that same exposure settings between formats and start to use the higher ISO's with lower noise levels this lower amplification of FF rewards you with, you'll find the true nature of the light gathering of FF lenses on FF sensors.
If you really think you can freely quote focal lengths in 35mm equivalent without quoting apertures in 35mm equivalents, then you're sounding very much like the Panasonic marketing department:

Surely you know that taking this small sensor, large aperture thing to extremes like this does not result in this Panasonic FZ200 having a lens equivalent of a 600mm f2.8 at full zoom, as Panasonic would like you to think? In terms of framing, yes. In terms of exposure due to ISO tweaking of the sensor, yes. In terms if DoF, no. In terms if light gathering, no! There's no way that a 'slow' Canon 600/4 II on a 1D X
as a package gathers less light than that 'f2.8' lens/sensor combo of the Panasonic. Yes, both at f4, 1/1000th of a sec, they'll both have to use an identical ISO to expose the same, but look past using the same rated ISO and guess which combo would work best to get clean images at high shutter speeds in low light?
If you can understand that, then surely you can see the 1.6x crop gives just over a stop less light gathering than a full frame sensor - making f1.8 on crop equal in light capturing terms to an aperture 1.6x smaller on FF - f2.88.