Uniform

Definition

A probability distribution where all values within a finite interval $[a,b]$ are equally likely. The density function is constant between the bounds. It is often denoted as $X\sim U(a,b)$.

Probability Density Function (PDF)

The height of the density function is determined by the requirement that the total area under the curve must equal 1.

$$f(x)=\{\begin{array}{ll}\frac{1}{b-a}& \text{for }a\le x\le b\\ 0& \text{otherwise}\end{array}$$

Cumulative Distribution Function (CDF)

The CDF increases linearly from 0 to 1 over the interval $[a,b]$.

$$F(x)=\{\begin{array}{ll}0& \text{for }x<a\\ \frac{x-a}{b-a}& \text{for }a\le x\le b\\ 1& \text{for }x>b\end{array}$$

Key Statistics

Expectation (Mean):

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

Variance:

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

Median:

$$\text{Median}=\frac{a+b}{2}$$

Moment Generating Function:

$$M_X(t)=\{\begin{array}{ll}\frac{e^{tb}-e^{ta}}{t(b-a)}& \text{for }t\ne 0\\ 1& \text{for }t=0\end{array}$$

Standard Uniform Distribution

A special case where $a=0$ and $b=1$.

$$f(x)=1\text{for }0\le x\le 1$$