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Encyclopedia > Uniform distribution (discrete)
Parameters Probability mass function n=5 where n=b-a+1 Cumulative distribution function  $a in (dots,-2,-1,0,1,2,dots),$ $b in (dots,-2,-1,0,1,2,dots),$ $n=b-a+1,$ $k in {a,a+1,dots,b-1,b},$ $begin{matrix} frac{1}{n} & mbox{for }ale k le b 0 & mbox{otherwise } end{matrix}$ $begin{matrix} 0 & mbox{for }kb end{matrix}$ $frac{a+b}{2},$ $frac{a+b}{2},$ N/A $frac{n^2-1}{12},$ $0,$ $-frac{6(n^2+1)}{5(n^2-1)},$ $ln(n),$ $frac{e^{at}-e^{(b+1)t}}{n(1-e^t)},$ $frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}$

If a random variable has any of n possible values $k_1,k_2,dots,k_n$ that are equally probable, then it has a discrete uniform distribution. The probability of any outcome ki  is 1 / n. A simple example of the discrete uniform distribution is throwing a fair dice. The possible values of k are 1, 2, 3, 4, 5, 6; and each time the dice is thrown, the probability of a given score is 1/6.

In case the values of a random variable with a discrete uniform distribution are real, it is possible to express the cumulative distribution function in terms of the degenerate distribution; thus In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... $F(k;a,b,n)={1over n}sum_{i=1}^n H(k-k_i)$

where the Heaviside step function H(xx0) is the CDF of the degenerate distribution centered at x0. This assumes that consistent conventions are used at the transition points. In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of half-open intervals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation. In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points. ... A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. ... Results from FactBites:

 Uniform distribution (520 words) The uniform distributions are simple probability distributions which, in the discrete case, can be characterized by saying that all possible values are equally probable. Although the uniform distribution is not commonly found in nature, it is particularly useful for sampling from arbitrary distributions. The normal distribution is an important example where the inverse transform method is not efficient.
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