Discrete Uniform Distribution

TL;DR

• Models situations with a finite set of discrete outcomes that are equally likely.
• Represented by a probability mass function assigning the same probability to each possible value.
• Commonly used for fair dice, simple lotteries, and other uniformly random discrete events.

Definition

The discrete uniform distribution is a probability distribution that assigns equal probability to each possible value within a set of mutually exclusive and exhaustive outcomes.

Explanation

Outcomes in a discrete uniform distribution are discrete (individual and separate) and mutually exclusive, and the probabilities are uniform (the same for every outcome). The distribution is described by a probability mass function that gives the probability of each possible outcome. The expected value (mean) of a discrete uniform distribution is the average of the possible outcomes. The variance is equal to the square of the range of possible outcomes divided by 12.

Examples

Fair six-sided die

The six faces are equally likely; each outcome has probability 1/6:

$$f(x) = 1/6, \text{ for } x = 1, 2, 3, 4, 5, 6$$

The expected value (mean) for this die is 3.5. The variance is given as 2.92, calculated as:

$$\frac{(6-1)^2}{12} = 2.92$$

Lottery with 10 possible numbers

Each number has equal probability 1/10 of being drawn:

$$f(x) = 1/10, \text{ for } x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10$$

Use cases

• Modeling outcomes of random events such as the roll of a die or the draw of a lottery number.
• Modeling distributions of certain types of data, for example test scores or the number of days it takes for a customer to return a product.

Related terms

• Probability mass function
• Expected value
• Variance