Uniform Distribution

What it is

A uniform distribution is a probability distribution in which every outcome within a specified set or interval is equally likely. It can be discrete (a finite set of outcomes) or continuous (an entire interval of values).

Key points

• All possible outcomes have equal probability.
• Discrete uniform: a finite number of distinct outcomes, each with probability 1/n.
• Continuous uniform: an infinite number of outcomes across an interval [a, b], each point in the interval equally likely in density terms.
• When plotted, a uniform distribution appears as a rectangle (constant height).

Discrete vs. continuous

Discrete uniform

• Example: rolling a fair six-sided die. Outcomes {1,2,3,4,5,6} each have probability 1/6.
• General formula: P(x) = 1/n for each of the n possible values.
• Expectation (mean) for values x1,…,xn: E[X] = (1/n) * sum(xi).

Continuous uniform

• Example: an ideal random number uniformly chosen between 0 and 1.
• Probability density function (pdf):
  f(x) = 1/(b − a) for a ≤ x ≤ b, and 0 otherwise.
• Mean: E[X] = (a + b)/2.
• Variance: Var(X) = (b − a)^2 / 12.

Visualizing a uniform distribution

• Discrete: bars of equal height for each outcome.
• Continuous: a flat horizontal line across the interval [a, b] (rectangular shape) representing constant density.
• Examples: drawing a suit from a well-shuffled deck (each suit = 1/4), flipping a fair coin (heads or tails = 1/2).

Example: cards

If you use a 40-card deck (only number cards, no jokers or face cards), each card has probability 1/40 of being drawn. The probability of drawing any heart is 10/40 = 1/4, because suits are equally represented.

Uniform vs. normal distribution

• Uniform: every value in the range is equally likely; shape is rectangular.
• Normal: values cluster around the mean in a bell-shaped curve; probabilities decrease as you move away from the mean.
• Both distributions integrate (area under the curve) to 1, but their shapes and implications for variability differ.

Simple explanation

With a uniform distribution, every allowed outcome is just as likely as any other. Rolling a fair die gives each face an equal chance.

Quick formulas

• Discrete uniform: P(x) = 1/n (n = number of outcomes).
• Continuous uniform on [a, b]: f(x) = 1/(b − a); E[X] = (a + b)/2; Var(X) = (b − a)^2/12.

Takeaway

Uniform distributions model situations where outcomes are equally likely across a finite set or continuous interval. They are a foundational, easy-to-interpret class of probability distributions used in simulations, random sampling, and theoretical analysis.