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Uniform distribution (discrete)

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discrete uniform
Probability mass function
DUniform distribution PDF
n=5 where n=b-a+1
Cumulative distribution function
DUniform distribution CDF
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 Gleichverteilungnl:Discrete uniforme verdeling fr:loi uniforme discrète

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