# 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 $a\in (\dots ,-2,-1,0,1,2,\dots )\,$ $b\in (\dots ,-2,-1,0,1,2,\dots )\,$ $n=b-a+1\,$ $k\in \{a,a+1,\dots ,b-1,b\}\,$ ${\begin{matrix}{\frac {1}{n}}&{\mbox{for }}a\leq k\leq b\ \\0&{\mbox{otherwise }}\end{matrix}}$ ${\begin{matrix}0&{\mbox{for }}kb\end{matrix}}$ ${\frac {a+b}{2}}\,$ ${\frac {a+b}{2}}\,$ N/A ${\frac {n^{2}-1}{12}}\,$ $0\,$ $-{\frac {6(n^{2}+1)}{5(n^{2}-1)}}\,$ $\ln(n)\,$ ${\frac {e^{at}-e^{(b+1)t}}{n(1-e^{t})}}\,$ ${\frac {e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}}$ 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. 