Uniform distribution (discrete)

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discrete uniform
Probability mass function n=5 where n=b-a+1
Cumulative distribution function
Parameters	$a \in (\dots,-2,-1,0,1,2,\dots)\,$ $b \in (\dots,-2,-1,0,1,2,\dots)\,$ $n=b-a+1\,$
Support	$k \in \{a,a+1,\dots,b-1,b\}\,$
Probability mass function (pmf)	$\begin{matrix} \frac{1}{n} & \mbox{for }a\le k \le b\ \\0 & \mbox{otherwise } \end{matrix}$
Cumulative distribution function (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	$\frac{a+b}{2}\,$
Mode	N/A
Variance	$\frac{n^2-1}{12}\,$
Skewness	$0\,$
Excess kurtosis	$-\frac{6(n^2+1)}{5(n^2-1)}\,$
Entropy	$\ln(n)\,$
Moment-generating function (mgf)	$\frac{e^{at}-e^{(b+1)t}}{n(1-e^t)}\,$
Characteristic function	$\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}\,$

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 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.

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Category: Discrete distributions

Uniform distribution (discrete)

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