Flavors of static: random distributions and noise

It's day one of noise generation in our flagship product! There are many ways to generate visual noise; our first algorithm is a simple static pixel generator. For these examples I'm generating a random integer from 0 to 255 and using that value to calculate a color on the gradient between two colors (here, blue and orange). Here's the initial output:

It's easiest to compare other output zoomed in a bit - here's a section zoomed at 1200%:

I'm interested in tweaking the actual noise generation, which here means changing the random value generation (not just applying visual adjustments to the output). Filtering the random unit through a log() function creates a curve that produces more values closer to the maximum and few values closer to the minimum - which interpreted as color, means we see more orange:

The complementary approach uses an exponential function to produce a distribution that increases more slowly at first - mapping more values to the lower end of the scale, here meaning more blue: