Uniform0a

Uniform0a is a discrete, finite uniform distribution defined on the set of integers {0, 1, ..., a}, where a is a nonnegative integer. In this convention each value in the range has equal probability, namely P(X = k) = 1/(a + 1) for k = 0, 1, ..., a. Uniform0a is a discrete analogue of the continuous Uniform(0, a) distribution and is sometimes used in contexts where a finite, equally likely set of outcomes is required.

Definition and notation: For each integer a ≥ 0, a random variable X is said to follow Uniform0a if it takes each integer in {0,...,a} with probability 1/(a+1). The distribution is symmetric around a/2, reflecting equal likelihood of all outcomes in the range.

Moments and distribution properties: The mean is E[X] = a/2, and the variance is Var(X) = a(a + 2)/12. The cumulative distribution function is F(x) = 0 for x < 0, F(x) = floor(x)/(a+1) for 0 ≤ x < a, and F(x) = 1 for x ≥ a. The probability mass function is p(k) = 1/(a+1) for k ∈ {0, ..., a}. The moment generating function is M_X(t) = (1/(a+1)) * sum_{k=0}^a e^{tk} = (1 - e^{t(a+1)}) / ((a+1)(1 - e^{t})) for t ≠ 0, with M_X(0) = 1.

Relations and limits: Uniform0a is a special case of the discrete uniform distribution on a finite set. As a grows large, the standardized variable (X − a/2) / sqrt(a(a+2)/12) converges in distribution to a standard normal distribution, reflecting a central limit type behavior for large ranges.

Applications: It is used in simple randomized algorithms, combinatorial modeling, hashing, and simulations where a finite, evenly distributed set of outcomes is needed without resorting to continuous models.