UNIFORM DISTRIBUTION (DISCRETE)

{{Probability distribution|
name =discrete uniform|
type =mass|
pdf_image =
n=5 where n=b-a+1|
cdf_image =
|
parameters =$a\; in\; (dots,-2,-1,0,1,2,dots),$
$b\; in\; (dots,-2,-1,0,1,2,dots),$
$n=b-a+1,$|
support =$k\; in\; \{a,a+1,dots,b-1,b\},$|
pdf =$$
egin{matrix}
rac{1}{n} & mbox{for }ale k le b \0 & mbox{otherwise }
end{matrix}
|
cdf =$$
egin{matrix}
0 & mbox{for }kb
end{matrix}
|
mean =|
median =|
mode =N/A|
variance =|
skewness =$0,$|
kurtosis =|
entropy =$ln(n),$|
mgf =|
char =|
}}
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.
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 $k\_i$  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
:$F(k;a,b,n)=\{1over\; n\}sum\_\{i=1\}^n\; H(k-k\_i)$
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.