Uniform Distribution
Definition
A continuous random variable $X \sim \text{Uniform}(a, b)$ has PDF
$$ f(x) = \frac{1}{b - a}, \quad a \leq x \leq b, $$and $f(x) = 0$ otherwise.
Moments
$$ \mathbb{E}[X] = \frac{a + b}{2}, \qquad \text{Var}(X) = \frac{(b - a)^2}{12}. $$Role in Sampling
The standard uniform $\text{Uniform}(0,1)$ is the foundation of all random number generation. It is used as the proposal distribution in Rejection Sampling.