Discrete Uniform Distribution
A discrete random variable $X$ is said to have a uniform distribution if its probability mass function (pmf) is given by
$P(X=x)=\frac{1}{N},;; x=1,2, \cdots, N$
Mean of Discrete Uniform Distribution
The expected value of discrete uniform random variable $X$ is
$E(X) =\frac{N+1}{2}$.
Variance of Discrete Uniform Distribution
The variance of discrete uniform random variable $X$ is
$V(X) = \dfrac{N^2-1}{12}$.
Moment generation function of discrete uniform distribution
$M_X(t) = \dfrac{e^t (1 - e^{tN}}{N (1 - e^t)}$.
General discrete uniform distribution
A general discrete uniform distribution has a probability mass function
$P(X=x)=\frac{1}{b-a+1},;; x=a,a+1,a+2, \cdots, b$
Distribution function of general discrete uniform random variable $X$ is
$F(x) = P(X\leq x)=\dfrac{x-a+1}{b-a+1}; a\leq x\leq b$
The expected value of above discrete uniform random variable $X$ is
$E(X) =\dfrac{a+b}{2}$.
The variance of above discrete uniform random variable $X$ is
