A discrete uniform random variable X with parameters a and b has probability mass function

f(x) = 1/ (b − a + 1) where x = a, a+1, a+2, ..., b

The cumulative distribution function is F(x) = P(X ≤ x) = (x−a+1)/ (b−a+1)

The survivor function of X is S(x) = P(X ≥ x) = (b−x+1)/ (b−a+1)

The hazard function of X is h(x) = f(x)/ S(x) = 1/ (b−x+1)

The inverse distribution function of X is F −1 (u) = a+[u(b−a+1)] 0 < u < 1.

The median, m, of X is m = (a+b)/ 2

The moment generating function of X is M(t) = E (etX) = (eat −e(b+1)t)/ {(b−a+1)(1−et)}

The population mean, variance, skewness, and kurtosis of X are

E[X] = (a+b)/ 2

V[X] = {(b−a+1)2 −1}/ 12

E [{(X −µ)/ σ}3] = 0

E [{(X −µ)/ σ} 4] = 6 [(b−a+1)2 +1)]/ 5 [(b−a+1)2 −1)]