Are the stars distributed in uniform distribution, on the celestial dome, with respect to brightness? Has there ever been a statistical analysis of the distribution of stars in the sky (on the surface of the celstial dome), by brightness? I want to know if they are uniformly distributed.

For example, all stars with Apparent Magnitude from 1.0 to 2.0. Are these uniformly distributed? Then all stars from 2.0 to 3.0, etc.

EDIT: To avoid bias for the Milky Way, there should be some cutoff magnitude. I don't know exactly what's the best cutoff, but I would guess somewhere around 4.0 to 5.0 apparent magnitude. If the cutoff is 4.0, then that gives us 3 tiers to look at (1 to 2, 2 to 3, 3 to 4). Are each of those tiers uniformly distributed?

Btw, the "obvious" distribution of theta from 0 to 2pi and phi from 0 to pi will NOT produce a uniform distribution. That will produce clustering near the poles. Wolfram Alpha explains it.)

I did google around for this. The results are all about star clusters or binaries, not the actual starfield of the night sky, or sometimes about how to program a random set of stars on the sphere.

That's not what I want. I'm looking for a statistical analysis on our own starfield. Has it ever been done before? What were the results?
Naked eye stars are not distributed uniformly in the sky. That is because the median naked eye star is at a distance of 440 light years, and this is far enough away that some of the details of Galactic structure start to become apparent. Most importantly, the density of stars increase towards the Galactic midplane and has a scale height of a few hundred light years and thus there is an overdensity of stars towards the Galactic plane.

I did an analysis of this in https://astronomy.stackexchange.com/a/10260/2531 , but the important plot is repeated below. It shows the normalised density of stars (per square degree) as a function of Galactic latitude. You can see the peak at low Galactic latitude and even work out that our viewing point is probably slightly above the Galactic plane, given that the peak is at about $-10^{\circ}$. The peak is still there in 3rd magnitude stars. You could repeat for a sample limited to very bright stars, but you run into number statistics trouble. 

The distribution of bright stars with Galactic latitude.


EDIT: Extra mile time. Below I show the Hipparcos data, divided into 4 apparent magnitude bins. Points are plotted on a RA vs Dec Aitoff projection. Superimposed in red is a Galactic coordinate grid. There are too few stars with $V&lt;2$ to form much of an opinion, but for fainter stars, a ring-like structure, coincident with the Galactic coordinates equator (i.e. Galactic latitude of zero, or just below, as I showed in the 1D projection above) is clearly seen.