Statistical Sampling

Continuous uniform distribution

A stochastic variable X follows a continuous uniform fistribution if it is continuous (i.e. it can take any value between two values a and b), and if the probability that is takes a value inside any subinterval (c, d) \in (a, b) equals \frac{d - c}{b - a}. Thus the probability fistribution function is a constant, namely f(x) = \frac{1}{b - a} everywhere (for all x \in (a, b)).

Basic properties:
Notation:
X \sim \mathrm{U}(a, b)
Type:
Continuous
Parameters:
a, b \in \mathbb{R}, \quad a < b
Variables:
X
Support:
[a, b]
PDF:
f(x) = \frac{1}{b - a}, \quad a \le x \le b
CDF:
F(x) = P(X \le x) = \frac{x - a}{b - a}
Mean:
\frac{a + b}{2}
Variance:
\frac{1}{12} (b - a)^2