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

f(x) = \frac{1}{b − a} a < x < b,

for −∞ < a < b < ∞. The uniform distribution is used to model a random variable that is equally likely to occur between a and b. The uniform distribution is central to random variate generation. The uniform distribution would be an appropriate model for the position of the puncture on a flat tire. Event times associated with a Poisson process are uniformly distributed over an interval. The probability density function is illustrated below.

PSfrag replacements

x f(x) a

b \frac{1}{b−a}

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{x − a}{b − a} a < x < b.

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{b − x}{b − a} a < x < b.

The hazard function on the support of X is

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

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = ln(b − a) − ln(b − x) a < x < b.

The inverse distribution function of X is

F^{−1} (u) = a + u(b − a) 0 < u < 1.

The median of X is

\frac{1}{2}(a + b).

The moment generating function of X is

M(t) = E [e^{tX}] = \{ \begin{matrix}1 & t = 0 \\ \frac{e^{bt}−e^{at}}{t(b−a)} & t ≠ 0. \end{matrix}

The characteristic function of X is

ϕ(t) = E [e^{itX}] = \{ \begin{matrix}1 & t = 0 \\ \frac{e^{ibt}−e^{iat}}{it(b−a)} & t ≠ 0. \end{matrix}

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

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

APPL verification: The APPL statements

X := UniformRV(a, b); CDF(X); SF(X); HF(X); CHF(X); IDF(X); MGF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the cumulative distribution, survivor function, hazard function, cumulative hazard function, inverse, moment generating function, population mean, variance, skewness, kurtosis.