Standard uniform distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ U(0,1) is used to indicate that the random variable X has the standard uniform distribution with minimum 0 and maximum 1. A standard uniform random variable X has probability density function

f(x) = 1 0 < x < 1.

The standard uniform distribution is central to random variate generation. The probability density function is illustrated below.

PSfrag replacements

x f(x) 0 1 1

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = x 0 < x < 1.

The survivor function on the support of X is

S(x) = P(X ≥ x) = 1 − x 0 < x < 1.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{1}{1 − x} 0 < x < 1.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = −ln(1 − x) 0 < x < 1.

The inverse distribution function of X is

F^{−1} (u) = u 0 < u < 1.

The median of X is 1/2.

The moment generating function of X is

M(t) = \{ \begin{matrix}1 & t = 0 \\ \frac{e^{t}−1}{t} & t ≠ 0 \end{matrix}

The characteristic function of X is

ϕ(t) = \{ \begin{matrix}1 & t = 0 \\ \frac{e^{it}−1}{it} & t ≠ 0 \end{matrix}

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

E[X] = \frac{1}{2}  V [X] = \frac{1}{12}  E [( \frac{X − μ}{σ} ) ^{3}] = 0  E [( \frac{X − μ}{σ} ) ^{4}] = \frac{9}{5}.

APPL verification: The APPL statements

X := UniformRV(0,1); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the population mean, variance, skewness, kurtosis, and moment generating function.