Discrete Uniform Distribution

The simplest scenario in probability: every outcome is equally likely.

The "Fair Die Roll" Distribution

The Discrete Uniform distribution describes a situation where there are a finite number of outcomes, and each outcome is equally likely to occur.

The most classic example is a single roll of a fair six-sided die. The possible outcomes are [1, 2, 3, 4, 5, 6], and the probability of rolling any one of these numbers is exactly 1/6. There is no bias towards any particular outcome.

Interactive Uniform Distribution

Adjust the number of possible outcomes (

n

) to see how the probability changes.

Number of Outcomes (n): 6

Mean (

\mu

): 3.50

Variance (

\sigma^2

): 2.92

Core Concepts

Probability Mass Function (PMF)

The PMF gives the probability of observing exactly one specific outcome, `k`.

P(X=k) = \frac{1}{n}

Since there are $n$ possible outcomes and each is equally likely, the probability of any single outcome $k$ occurring is simply $1/n$ . For a 6-sided die, this is 1/6 for each face. For a 20-sided die, it's 1/20.

Mean (

\mu

): 3.50

Variance (

\sigma^2

): 2.92

Expected Value & Variance

Expected Value (Mean)

E[X] = \frac{n+1}{2}

For a fair die with outcomes 1 to `n`, the expected value is the average of the first and last outcome. For a 6-sided die, this is (1+6)/2 = 3.5. This is the balancing point of the distribution.

Variance

Var(X) = \frac{n^2 - 1}{12}

The variance measures the spread of the outcomes.

Applications

Quantitative Finance: Monte Carlo Simulation Assumptions

The Discrete Uniform distribution is often the starting point for Monte Carlo simulations. When modeling a decision with several equally likely strategic choices (e.g., "aggressively buy," "hold," "aggressively sell"), a quant might assign a uniform probability to each choice to simulate a trader's behavior under uncertainty before layering on more complex assumptions.