Difference between Continuous and Discrete Uniform Distribution

Last Updated : 23 Jul, 2025

Continuous and discrete uniform distributions are two types of probability distributions. A continuous uniform distribution has an interval of equally likely values. Instead, a discrete uniform distribution applies to a finite set of outcomes with equal probabilities.

Understanding the difference between continuous and discrete uniform distribution is crucial for anyone studying probability and statistics. These two types of distributions represent data differently, with continuous uniform distribution describing outcomes over a continuous range and discrete uniform distribution dealing with distinct, separate values.

In this article, we will discuss continuous and discrete uniform distribution along with a difference between them.

Table of Content

What is Continuous Distribution?
What is Discrete Uniform Distribution?
Formula of Continuous and Discrete Uniform Distribution
Difference between Continuous and Discrete Uniform Distribution

What is Continuous Distribution?

Continuous uniform distribution is a probability distribution in which all outcomes are equally likely within a specified interval [a,b]. In other words, the probability density function (PDF) is constant over this interval, and the distribution is defined by the two parameters a and b, which are the lower and upper bounds, respectively.

In a continuous distribution:

The total area under the probability density function (PDF) curve for a continuous distribution is equal to 1.
Examples of continuous distributions include normal, uniform, and exponential distributions.

Example of Continuous Distribution

Examples of continuous uniform distribution are:

Random Time of Day
Random Position on a Line Segment
Random Point in a Unit Square

Properties of Continuous Uniform Distribution

The properties of continuous uniform distribution are:

Symmetry: The distribution is perfectly symmetric about the mean.

Bell-shaped Curve: The shape of the normal distribution is often referred to as a "bell curve".

Mean, Median, and Mode: In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution.

What is Discrete Uniform Distribution?

Discrete uniform distribution is a kind of probability distribution in which every possible result has equal likelihood of occurrence. When there are limited possibilities and every one of them is equally likely, this distribution is applied.

Example of Discrete Uniform Distribution

Examples of discrete uniform distribution are:

Rolling a Fair Die
Drawing a Card from a Deck
Choosing a Random Day of the Week
Flipping a fair coin

Properties of Discrete Uniform Distribution

The properties of discrete uniform distribution are:

Equal Probability: Each outcome has an equal chance of occurring.

Finite Set of Outcomes: The possible outcomes are discrete and finite.

Uniformity: The probability mass function (PMF) is constant over the range of possible outcomes.

Formula of Continuous and Discrete Uniform Distribution

Below are formulas of continuous and discrete uniform distribution:

Distribution Type	Description	Probability Density Function/ Probability Mass Function	Cumulative Distribution Function
Discrete Uniform	Finite set of equally likely outcomes	P(X=x)= 1/n for x=x₁,x₂,...,x_n	F(x)=Number of outcomes ≤ x/n
Continuous Uniform	Continuous range of equally likely outcomes between a and b	f(x)= 1/(b-a) for a ≤ x ≤ b	F(x)= \begin{cases} 0 & \text{if } x < a \\ \frac{x - a}{b - a} & \text{if } a \leq x \leq b \\ 1 & \text{if } x > b \end{cases}