Uniform distribution (discrete)

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	<math>a \in (\dots,-2,-1,0,1,2,\dots)\,</math> <math>b \in (\dots,-2,-1,0,1,2,\dots)\,</math> <math>n=b-a+1\,</math>
Support	<math>k \in \{a,a+1,\dots,b-1,b\}\,</math>
Probability mass function (pmf)	<math> \begin{matrix} \frac{1}{n} & \mbox{for }a\le k \le b\ \\0 & \mbox{otherwise } \end{matrix} </math>
Cumulative distribution function (cdf)	<math> \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} </math>
Mean	<math>\frac{a+b}{2}\,</math>
Median	<math>\frac{a+b}{2}\,</math>
Mode	N/A
Variance	<math>\frac{n^2-1}{12}\,</math>
Skewness	<math>0\,</math>
Excess kurtosis	<math>-\frac{6(n^2+1)}{5(n^2-1)}\,</math>
Entropy	<math>\ln(n)\,</math>
Moment-generating function (mgf)	<math>\frac{e^{at}-e^{(b+1)t
Characteristic function	{{{char}}}

{n(1-e^t)}\,</math>|

 char       =<math>\frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}</math>|

}}

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 <math>n</math> possible values <math>k_1,k_2,\dots,k_n</math> that are equally probable, then it has a discrete uniform distribution. The probability of any outcome <math>k_i</math>  is <math>1/n</math>. A simple example of the discrete uniform distribution is throwing a fair dice. The possible values of <math>k</math> 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

<math>F(k;a,b,n)={1\over n}\sum_{i=1}^n H(k-k_i)</math>

where the Heaviside step function <math>H(x-x_0)</math> is the CDF of the degenerate distribution centered at <math>x_0</math>. 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.

cs:Rovnoměrné rozdělení