# Uniform distribution (discrete)

 Parameters Probability mass functionDiscrete uniform probability mass function for n=5n=5 where n=b-a+1 Cumulative distribution functionDiscrete uniform cumulative density function for n=5 $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 }a\le 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 $\frac{n^2-1}{12}\,$ $0\,$ $-\frac{6(n^2+1)}{5(n^2-1)}\,$ $\ln(n)\,$ $\frac{e^{at}-e^{(b+1)t {{{char}}} {n(1-e^t)}\,$|
 char       =$\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}$|


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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 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

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

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.

cs:Rovnoměrné rozdělení