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Uniform Distribution

Uniform Distribution

Definition

Core Statement

The Uniform Distribution (Continuous) assigns equal probability to all values in a specified interval [a,b]. Every outcome in the range is equally likely. It is the "flat" distribution.

Purpose

  1. Model scenarios where all outcomes are equally likely.
  2. Generate random numbers for simulations (basis of random number generators).
  3. Serve as a non-informative prior in Bayesian Statistics.
  4. Baseline for comparing other distributions.

When to Use

Use Uniform Distribution When...

  • All values in a range are equally probable.
  • No information suggests one value is more likely than another.
  • Generating random samples for Monte Carlo simulations.

Theoretical Background

Notation

X∼Uniform(a,b)

where a is the minimum and b is the maximum.

Probability Density Function (PDF)

f(x|a,b)={1b−aif a≤x≤b0otherwise

Constant density across the interval.

Cumulative Distribution Function (CDF)

F(x|a,b)={0if x<ax−ab−aif a≤x≤b1if x>b

Properties

Property Formula
Mean μ=a+b2 (midpoint)
Variance σ2=(b−a)212
Median a+b2 (same as mean)
Mode Any value in [a,b] (all equally likely)

Standard Uniform: U(0,1)

Special case where a=0,b=1. Used as the foundation for generating all other random variables via inverse transform sampling.

Worked Example: Waiting for a Train

Problem

A commuter train arrives every 15 minutes. You arrive at the station at a random time, so your waiting time X is uniformly distributed between 0 and 15 minutes.

Questions:

  1. What is the probability you wait less than 5 minutes?
  2. What is the average (expected) waiting time?

Solution:

Parameters: a=0, b=15. Distribution X∼U(0,15).
PDF height = 115−0=115.

1. Probability wait < 5 mins (P(X<5)):

P(X<5)=Base×Height=(5−0)×115P(X<5)=515=13≈0.333

Result: ~33.3% chance of a short wait.

2. Average Waiting Time (E[X]):

μ=a+b2=0+152=7.5 minutes

Result: On average, you will wait 7.5 minutes.

Assumptions

The Uniform distribution is a model choice:

Limitations

Pitfalls

  1. The "Lazy Prior" Fallacy: Assuming a distribution is uniform just because you have no data can be dangerous (the "Principle of Indifference"). Sometimes reality is bell-shaped or power-law.
  2. Pseudo-randomness: Computer "uniform" generators are deterministic algorithms. For cryptography, you need cryptographically secure RNGs.
  3. Boundary Bias: Real-world metrics rarely have hard "walls" like a and b with zero probability outside.

Python Implementation

from scipy.stats import uniform
import numpy as np
import matplotlib.pyplot as plt

# Uniform on [2, 8]
a, b = 2, 8
dist = uniform(loc=a, scale=b-a)  # scipy uses loc=a, scale=b-a

# Mean and Variance
print(f"Mean: {dist.mean():.2f}")
print(f"Variance: {dist.var():.2f}")

# P(3 < X < 6)
prob = dist.cdf(6) - dist.cdf(3)
print(f"P(3 < X < 6): {prob:.4f}")

# Visualize PDF
x = np.linspace(0, 10, 500)
plt.plot(x, dist.pdf(x), lw=3, label=f'Uniform({a}, {b})')
plt.xlabel('x')
plt.ylabel('Density')
plt.title('Uniform Distribution')
plt.legend()
plt.grid(alpha=0.3)
plt.show()

# Generate Random Sample
sample = dist.rvs(size=1000)
plt.hist(sample, bins=30, density=True, alpha=0.6, edgecolor='black')
plt.title('Histogram of 1000 Uniform Samples')
plt.show()

R Implementation

# Uniform on [2, 8]
a <- 2
b <- 8

# Mean
(a + b) / 2

# P(3 < X < 6)
punif(6, min = a, max = b) - punif(3, min = a, max = b)

# Visualize PDF
curve(dunif(x, min = a, max = b), from = 0, to = 10, lwd = 3,
      xlab = "x", ylab = "Density",
      main = paste("Uniform(", a, ", ", b, ")", sep=""), col = "blue")

# Random Sample
runif(10, min = a, max = b)

Interpretation Guide

Scenario Interpretation
Scenario Interpretation
---------- ----------------
U(0,1) Standard reference. If p-values are uniform, H0 is true.
Mean vs Median In Uniform, Mean = Median. Symmetry holds.
Variance Depends heavily on the range width (b−a). σ∝(b−a).
Constant PDF "Flat" likelihood. Every value is equally surprising (or unsurprising).