the continuous uniform distribution

Imagine a bus that you know will arrive sometime in the next ten minutes, but you have absolutely no information about when — any instant in that window is as likely as any other. That is the continuous uniform distribution: a random number that is equally happy to land anywhere on a fixed interval, with no spot preferred over its neighbours.

Write X ~ Uniform(a, b). The density is flat: f(x) = 1/(b - a) for x between a and b, and 0 outside. The height 1/(b - a) is whatever it must be so the area under the line equals 1 — over a width of (b - a), a rectangle of that height has area exactly 1. Probabilities are just lengths: P(c < X < d) = (d - c)/(b - a), the fraction of the interval that the sub-window covers. The mean sits in the middle, E[X] = (a + b)/2, and the spread is Var(X) = (b - a)^2 / 12. The cdf rises in a straight ramp from 0 at a to 1 at b.

The uniform is the simplest continuous model and the seed of all the others: a computer's random number generator produces (approximately) Uniform(0, 1) numbers, and the inverse-transform method turns those into samples from any distribution you like. One honest caution — 'uniform' here means uniform on an interval; people sometimes wrongly assume any 'random' continuous quantity is uniform, but most real measurements (heights, errors, waiting times) are not flat at all.