Uniform distribution (discrete)

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discrete uniform
Probability mass function
Discrete uniform probability mass function for n=5
n=5 where n=b-a+1
Cumulative distribution function
Discrete uniform cumulative density function for n=5

Probability mass function (pmf)
Cumulative distribution function (cdf)
Mode N/A
Excess kurtosis
Moment-generating function (mgf)
Characteristic function

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 possible values that are equally probable, then it has a discrete uniform distribution. The probability of any outcome   is . A simple example of the discrete uniform distribution is throwing a fair dice. The possible values of 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

where the Heaviside step function is the CDF of the degenerate distribution centered at . 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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