Uniform Distribution (Discrete)

Core Concepts

If a random variable $X$ follows a Discrete Uniform Distribution between a minimum integer $a$ and a maximum integer $b$, we write:

Where:

• $a$ is the minimum possible integer value.
• $b$ is the maximum possible integer value.
• $n = b - a + 1$ is the total number of possible outcomes.

Probability Mass Function (PMF)

Because every outcome is equally likely, the probability of any specific outcome $x$ is simply $1$ divided by the total number of possible outcomes ($n$).

Cumulative Distribution Function (CDF)

The probability that the outcome is less than or equal to $x$ grows linearly step-by-step:

Key Metrics

• Expected Value (Mean): $E[X] = \frac{a + b}{2}$ (Exactly in the middle)
• Variance: $\text{Var}(X) = \frac{n^2 - 1}{12}$

Real-World Examples

1. Rolling a Fair Die: A standard 6-sided die follows $\mathcal{U}\{1, 6\}$. The probability of rolling any specific number is exactly $\frac{1}{6}$.
2. Drawing a Card: Drawing a single card from a well-shuffled standard deck of 52 cards (ignoring suits).
3. Lottery Balls: Selecting a numbered ping-pong ball from a rotating lottery drum.

Interactive Visualization

Use the sliders below to adjust the minimum ($a$) and maximum ($b$) possible integer outcomes. Notice how the discrete bars always remain the exact same height as each other, and that height automatically adjusts so the sum of all bars perfectly equals 1.0.

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