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
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{\lfloor k \rfloor -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 ki  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 − x0) is the CDF of the degenerate distribution centered at x0. 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.

[edit] See also

v • d • e
Probability distributions
 
Discrete univariate with finite support
Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot