Equally likely values

One simple family of discrete distributions that is occasionally encountered is the family of uniform distributions. A uniform distribution arises when it can be argued that each integer value between two limits has the same chance of being observed — equally likely outcomes.

Definition

If a discrete random variable, \(X\), can only take integer values from \(a\) to \(b\), where \(a\) and \(b\) are integer constants, and each such value is equally likely, it is said to have a discrete uniform distribution.

\[ X \;\; \sim \; \; \UniformDistn(a, b) \]

Probability function

Since there are \((b-a+1)\) equally likely values between \(a\) and \(b\) (inclusive), the probability function of \(X\) is

\[ p(x) = \begin {cases} \displaystyle \frac 1 {b-a+1} & \quad \text{if } a \le x \le b \\[0.5em] 0 & \quad \text{otherwise} \end {cases} \]

Example

Rolling a 6-sided die
In board games, a 6-sided die is often rolled to determine the number of squares that a player moves in the next turn. Since each value from 1 to 6 is equally likely, the number of moves is
\[ X \;\; \sim \; \; \UniformDistn(1, 6) \]
and has probability function
\[ p(x) = \begin {cases} \frac 1 6 & \text{if } 1 \le x \le 6 \\[0.5em] 0 & \text{otherwise} \end {cases} \]

If the probabilities are displayed graphically in a bar chart, each bar has the same height.