The continuous uniform distribution represents a situation where all outcomes in a range between a minimum and maximum value are equally likely.
From a theoretical perspective, this distribution is a key one in risk analysis; many Monte Carlo software algorithms use a sample from this distribution (between zero and one) to generate random samples from other distributions (by inversion of the cumulative form of the respective distribution).
On the other hand, there are only a few real-life processes that have this form of uncertainty. These could include for example: the position of a particular air molecule in a room, the point on a car tire where the next puncture will occur, the number of seconds past the minute that the current time is, or the length of time that one may have to wait for a train. In oil exploration, the position of the oil-water contact in a potential prospect is also often considered to be uniformly continuously distributed.
For the distribution to apply to each situation, implied assumptions need to hold, and it is the validity of these assumptions that can be questioned. In the example concerning the waiting time for a train, one would need to assume that trains arrive in regular intervals but that we have no knowledge of the current time, not of other indicators (sound, wind) that a train is in the process of arriving. For this reason, the distribution is sometimes called the “no knowledge” distribution. One of the reasons that such a distribution is not of frequent occurrence in the natural world is that in many cases it is readily possible to establish more knowledge of a situation, and that in particular there is usually a base case or most likely value that can be estimated.