uniform distribution
The uniform distribution is the picture of total impartiality: every value in a range is equally likely, with no part favoured over another. Picture a perfectly balanced spinner that can stop anywhere on a dial between 0 and 1 — no spot is preferred. Its density is a flat rectangle, which is why it is sometimes called the rectangular distribution. It is the mathematical expression of 'I know the range but have no reason to prefer any value within it'.
The continuous uniform on the interval from a to b has constant density 1/(b-a) over that interval and zero outside it. Its mean sits exactly in the middle, (a+b)/2, and its variance is (b-a)^2/12. There is also a discrete uniform — a fair die is the classic case, with each of {1,2,3,4,5,6} equally likely. The flat shape makes the uniform the simplest non-trivial continuous distribution and a natural baseline.
The uniform earns its keep two ways in actuarial work. First, it is the engine of simulation: computers generate uniform random numbers and then transform them (via the inverse CDF) into draws from any distribution you like, the basis of all Monte Carlo modeling of losses and capital. Second, it appears as a modeling assumption — the Uniform Distribution of Deaths (UDD) assumption spreads deaths evenly across each year of age to interpolate a life table at fractional ages. As a loss model on its own, though, it is rarely realistic, since real losses are almost never equally likely across their whole range.
To simulate a claim from any loss distribution, a computer first draws U uniformly between 0 and 1, then sets the loss equal to the value whose CDF equals U. This inverse-transform trick turns flat randomness into any shape you need.
Uniform random numbers are the raw fuel of all simulation.
A uniform is rarely a credible model for actual losses — its main jobs are as a simulation building block and as a convenient interpolation assumption, not as a description of real-world risk.