Statistical Sampling

Discrete uniform distribution

Assume that a stochastic variable A can take any one out of a discrete set of values. If the probability that A takes any value is the same for all values, then A is said to follow a discrete uniform distribution. A classic example would be that of shaking a bag of marbles, each with a distinct colour, and then picking out one of them from the bag. Then the probability of picking any one colour is the same for all the colours, equal to \frac{1}{N}, where N is the number of distinct colours to pick from.

Basic properties:
Notation:
N \sim \mathrm{U}\{a, b\}
Type:
Discrete
Parameters:
a, b \in \mathbb{Z}, \quad a \le b
Variables:
N
Support:
\{ a, ..., b \}
PDF:
P(N = n) = \frac{1}{b - a + 1}, \quad a \le n \le b
CDF:
P(N \le n) = \frac{n - a + 1}{b - a + 1}
Mean:
\frac{a + b}{2}
Variance:
\frac{1}{12} \big( (b - a + 1)^2 - 1 \big)