Statistics/Distributions/Uniform

Continuous Uniform Distribution

Template:Probability distribution The (continuous) uniform distribution, as its name suggests, is a distribution with probability densities that are the same at each point in an interval. In casual terms, the uniform distribution shapes like a rectangle.

Mathematically speaking, the probability density function of the uniform distribution is defined as

$f:[a,b]\to ℝ$

$f\left(x\right)=\frac{1}{b-a}$

And the cumulative distribution function is:

$F\left(x\right)=\{\begin{array}{cc}0,& \text{if }x\le a\\ \frac{x-a}{b-a},& \text{if }a<x<b\\ 1,& \text{if }x\ge b\end{array}$

Mean

We derive the mean as follows.

$\mathrm{E}[X]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}xf(x)dx$

As the uniform distribution is 0 everywhere but [a, b] we can restrict ourselves that interval

$\mathrm{E}[X]=\underset{a}{\overset{b}{\int }}\frac{1}{b-a}xdx$

$\mathrm{E}[X]={\frac{1}{(b-a)}\frac{1}{2}{x}^2|}_a^b$

$\mathrm{E}[X]=\frac{1}{2(b-a)}[b^2-a^2]$

$\mathrm{E}[X]=\frac{b+a}{2}$

Variance

We use the following formula for the variance.

$\mathrm{Var}(X)=\mathrm{E}[X^2]-(\mathrm{E}[X]{)}^2$

$\mathrm{Var}(X)=[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)\cdot x^{2}dx]-{\left(\frac{b+a}{2}\right)}^2$

$\mathrm{Var}(X)=\left[\underset{a}{\overset{b}{\int }}\frac{1}{b-a}{x}^{2}dx\right]-\frac{(b+a{)}^2}{4}$

$\mathrm{Var}(X)={\frac{1}{b-a}\frac{1}{3}{x}^3|}_a^b-\frac{(b+a{)}^2}{4}$

$\mathrm{Var}(X)=\frac{1}{3(b-a)}[b^3-a^3]-\frac{(b+a{)}^2}{4}$

$\mathrm{Var}(X)=\frac{4(b^3-a^3)-3(b+a{)}^2(b-a)}{12(b-a)}$

$\mathrm{Var}(X)=\frac{(b-a{)}^3}{12(b-a)}$

$\mathrm{Var}(X)=\frac{(b-a{)}^2}{12}$

External links

• Interactive Uniform Distribution Web Applet (Java)