Uniform Random Variable: A Complete and Detailed Guide

Introduction

The uniform random variable is one of the simplest and most fundamental probability distributions. It models a situation in which all outcomes in a given interval are equally likely. It’s often used as a building block for other distributions and is essential in simulations and Monte Carlo methods.

Definition

A uniform random variable can be either discrete or continuous. The most common and widely used is the continuous uniform distribution.

A continuous uniform random variable X is said to be uniformly distributed over the interval [a, b], written as:

X ~ U(a, b)

This means that the probability of X falling anywhere in the interval [a, b] is equally likely.

Probability Density Function (PDF)

The PDF of a uniform distribution on [a, b] is defined as:

f(x) = 1 / (b - a),  for a ≤ x ≤ b

f(x) = 0,           otherwise

This flat or constant density function indicates that all values in the interval [a, b] are equally probable.

Cumulative Distribution Function (CDF)

The CDF, which gives the probability that X is less than or equal to a certain value x, is:

F(x) = 0               if x < a

F(x) = (x - a) / (b - a) if a ≤ x ≤ b F(x) = 1                  if x > b

Mean and Variance

For X ~ U(a, b):

• Mean: E[X] = (a + b) / 2
• Variance: Var(X) = (b - a)2 / 12

Example

Suppose a bus arrives at a stop every 20 minutes. If you arrive at a random time, your waiting time X is uniformly distributed between 0 and 20 minutes: X ~ U(0, 20).

– The probability that you wait less than 5 minutes:

P(X ≤ 5) = (5 - 0) / (20 - 0) = 5 / 20 = 0.25

– The average wait time:

E[X] = (0 + 20) / 2 = 10 minutes

Discrete Uniform Distribution

A discrete uniform distribution assigns equal probability to a finite set of n outcomes. For example, rolling a fair 6-sided die produces outcomes {1, 2, 3, 4, 5, 6}, each with probability 1/6.

If X is uniformly distributed over {x1, x2, ..., xn}, then:

P(X = xi) = 1 / n  for all i

Applications

• Random number generation
• Simulation and modeling in Monte Carlo methods
• Estimating probabilities in uniformly distributed data
• Game theory and lotteries

Python Code Example

from scipy.stats import uniform

Continuous uniform from a = 0 to b = 20

mean, var = uniform.mean(loc=0, scale=20), uniform.var(loc=0, scale=20) print("Mean:", mean) print("Variance:", var)

Probability of waiting less than 5 minutes

p = uniform.cdf(5, loc=0, scale=20) print("P(X ≤ 5):", p)

Conclusion

The uniform random variable is a foundational concept in probability and statistics. It models situations with equal likelihood and serves as a simple but powerful tool in data analysis and simulation. Whether dealing with continuous ranges or finite discrete sets, understanding uniform distributions helps establish deeper intuition for randomness and fair systems.