GleichverteilungsEigenschaft

Gleichverteilung, in probability theory, refers to a probability distribution in which all outcomes in a specified range are equally likely. In English, it is called the uniform distribution. There are two common forms: discrete and continuous.

In the discrete case, if X follows a uniform distribution on the integers a, a+1, ..., b, then n = b−a+1 and P(X = k) = 1/n for each k in {a, a+1, ..., b}. Each outcome has the same probability, and the distribution is finite and symmetric around (a+b)/2.

In the continuous case, X follows a uniform distribution on the interval [a, b] with a < b. The probability density function is f(x) = 1/(b−a) for x in [a, b], and 0 otherwise. The cumulative distribution function is F(x) = 0 for x < a, F(x) = (x−a)/(b−a) for a ≤ x ≤ b, and F(x) = 1 for x > b.

Moments: for both forms, the mean is (a+b)/2. The variance is (b−a)^2/12 in the continuous case, and ((b−a+1)^2 − 1)/12 in the discrete case.

Generation and estimation: to sample from a discrete uniform, draw a random integer in {a, ..., b}. To sample from a continuous uniform, use a + (b−a)·U where U ~ U(0,1). Endpoints can be estimated by observed minima and maxima, though this introduces bias in finite samples.

Applications include Monte Carlo simulations, randomized algorithms, and modeling ignorance or lack of prior information. The Gleichverteilung is a foundational, simple model with equal likelihood across a defined range.