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Uniform distribution (discrete)

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**discrete uniform**
Probability mass function n=5 where n=b-a+1
Cumulative distribution function n=5 where n=b-a+1. The convention is used that the cumulative mass function $F k (k i)$ is the probability that $k > = k i$
Parameters	$a \in (...,-2,-1,0,1,2,...)\,$ $b \in (...,-2,-1,0,1,2,...)\,$ $n=b-a+1\,$
Support	$k \in \{a,a+1,...,b-1,b\}\,$
Template:Probability distribution/link mass	$\begin{matrix} \frac{1}{n} & \mbox{for }a\le k \le b\ \\0 & \mbox{otherwise } \end{matrix}$
cdf	$\begin{matrix} 0 & \mbox{for }k<a\\ \frac{k-a+1}{n} & \mbox{for }a \le k \le b \\1 & \mbox{for }k>b \end{matrix}$
Mean	$\frac{a+b}{2}\,$
Median	$a+n/2\,$
Mode	N/A
Variance	$\frac{n^2-1}{12}\,$
Skewness	$0\,$
Kurtosis	$-\frac{6(n^2+1)}{5(n^2-1)}\,$
Entropy	$\ln(n)\,$
mgf	$e^{-at}\sum_{k=0}^{n-1}e^{kt}\,$
Char. func.	$e^{-iat}\sum_{k=0}^{n-1}e^{ikt}\,$

In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable.

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

$F(k;a,b,n)={1\over n}\sum_{i=1}^n H(k-k_i)$

where the Heaviside step function $H (x - x 0)$ is the CDF of the degenerate distribution centered at $x 0$ . This assumes that consistent conventions are used at the transition points.

See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation.de:Diskrete Gleichverteilung nl:Discrete uniforme verdeling fr:loi uniforme discrète

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