Uniform

Definition

Concept

Uniform distribution: all outcomes equally likely over defined support. Simplest continuous or discrete model. Basis for modeling ignorance or equal preference.

Formal Definition

Continuous uniform on interval [a, b]: PDF constant on [a, b], zero elsewhere. Discrete uniform on finite set: equal probability assigned to each element.

Support

Continuous case: support is bounded interval [a, b]. Discrete case: finite set of distinct values.

Types of Uniform Distribution

Continuous Uniform Distribution

Defined on real interval [a, b]. Infinite outcomes. Constant density: 1/(b-a).

Discrete Uniform Distribution

Defined on finite set {x1, x2, ..., xn}. Probability mass function (PMF): 1/n for each xi.

Generalizations

Multivariate uniform on geometric shapes. Circular uniform, spherical uniform as special cases.

Probability Density Function (PDF)

Continuous Case

Density constant on interval [a, b], zero elsewhere.

f(x) = { 1/(b - a), for a ≤ x ≤ b 0, otherwise}

Discrete Case

PMF equal probability 1/n for each of n outcomes.

P(X = x_i) = 1/n, i = 1, 2, ..., n

Normalization

Integral or sum over support equals 1. Ensures valid probability distribution.

Cumulative Distribution Function (CDF)

Continuous Uniform CDF

Linear increase from 0 at a to 1 at b.

F(x) = { 0, x < a (x - a)/(b - a), a ≤ x ≤ b 1, x > b}

Discrete Uniform CDF

Stepwise increase by 1/n at each discrete point.

Properties

Monotone non-decreasing, right-continuous, limits 0 and 1 at support boundaries.

Moments: Expectation and Variance

Mean (Expected Value)

Continuous uniform mean: midpoint of interval.

E(X) = (a + b) / 2

Variance

Variance measures spread; for continuous uniform:

Var(X) = (b - a)^2 / 12

Higher Moments

Skewness = 0 (symmetric), kurtosis = -6/5 (platykurtic).

Properties

Symmetry

Symmetric about mean (a+b)/2. No skewness.

Maximum Entropy

Among continuous distributions on [a,b], uniform has highest entropy.

Memorylessness

Uniform distribution is not memoryless.

Support Boundedness

Support strictly bounded; no tails.

Parameter Estimation

Method of Moments

Estimate a, b from sample mean and variance:

a = mean - sqrt(3) * std_devb = mean + sqrt(3) * std_dev

Maximum Likelihood Estimation (MLE)

MLE for continuous uniform: a_hat = min sample, b_hat = max sample.

Confidence Intervals

Constructed using order statistics; exact intervals derivable.

Applications

Random Number Generation

Core distribution for pseudorandom number generators; basis for inversion method.

Modeling Ignorance

Represents equal likelihood when no preference or information exists.

Simulation

Used for Monte Carlo methods, bootstrapping, and stochastic modeling.

Game Theory

Modeling equiprobable strategies or outcomes.

Simulation and Random Number Generation

Inverse Transform Sampling

Generate uniform U~U(0,1), then transform:

X = a + (b - a) * U

Algorithmic Implementation

Simple, efficient: direct scaling of uniform(0,1) output.

Quality Considerations

Depends on quality of base uniform generator; uniformity critical.

Relations to Other Distributions

Beta Distribution

Uniform is Beta(1,1) on [0,1].

Triangular Distribution

Triangular arises from convolution of two uniform variables.

Exponential Distribution

Uniform is not memoryless unlike exponential.

Discrete Uniform as Special Case

Discrete uniform is limit of multinomial with equal probabilities.

Multivariate Uniform Distribution

Definition

Equal probability over multi-dimensional bounded region (e.g., hypercube).

Joint PDF

Constant over support volume, zero outside.

Independence

Components independent if joint uniform is product of marginals.

Applications

Spatial models, sampling in high-dimensional spaces.

Limitations

Lack of Flexibility

Fixed shape; cannot model skewness, heavy tails, or multimodality.

Strict Support Bounds

Inappropriate for unbounded or semi-bounded data.

Not Memoryless

Unlike some distributions, uniform does not satisfy memoryless property.

Parameter Sensitivity

Estimation sensitive to outliers; boundary estimates can be biased.

References

• Casella, G., Berger, R. L., Statistical Inference, Duxbury, 2002, pp. 50-65.
• Ross, S. M., Introduction to Probability Models, 11th Ed., Academic Press, 2014, pp. 60-75.
• Devroye, L., Non-Uniform Random Variate Generation, Springer, 1986, pp. 45-55.
• Hogg, R. V., Tanis, E. A., Probability and Statistical Inference, 9th Ed., Pearson, 2014, pp. 120-130.
• Johnson, N. L., Kotz, S., Balakrishnan, N., Continuous Univariate Distributions, Vol. 1, Wiley, 1995, pp. 33-40.